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14580AK_FM.pgs 3/26/09 12:07 PM Page i Answer Key ALGEBRA 2 and TRIGONOMETRY AMSCO AMSCO SCHOOL PUBLICATIONS, INC. 315 HUDSON STREET, NEW YORK, N.Y. 10013 14580AK_FM.pgs 3/26/09 12:07 PM Page ii Please visit our Web site at: www.amscopub.com When ordering this book, please specify: N 159 K or ANSWER KEY/ALGEBRA 2 AND TRIGONOMETRY Copyright © 2009 by Amsco School Publications, Inc. No part of this Answer Key may be reproduced in any form without written permission from the publisher except by those teachers using the AMSCO textbook ALGEBRA 2 AND TRIGONOMETRY, who may reproduce or adapt portions of this key in limited quantities for classroom use only. Printed in the United States of America 1 2 3 4 5 6 7 8 9 10 14 13 12 11 10 09 14580FM.pgs 3/26/09 12:11 PM Page iv Contents Answer Keys For Enrichment Activities 246 For Extended Tasks 255 For Suggested Test Items 261 For SAT Preparation Exercises 269 For Textbook Exercises iv Chapter 1 271 Chapter 2 274 Chapter 3 277 Chapter 4 282 Chapter 5 291 Chapter 6 299 Chapter 7 303 Chapter 8 308 Chapter 9 312 Chapter 10 319 Chapter 11 324 Chapter 12 334 Chapter 13 343 Chapter 14 345 Chapter 15 349 Chapter 16 359 14580AKEA.pgs 3/26/09 12:07 PM Page 246 Answers for Enrichment Exercises Enrichment Activity 1-5: On the Ins and Outs 7. 1. a. 110 2 108 5 2 b. 380 2 378 5 2 c. (30 1 1)(30 1 2) 2 f30(30 1 3)g 5 302 1 3(30) 1 2 2 302 2 3(30) 52 d. (x 1 1)(x 1 2) 2 x(x 1 3) 5 x2 1 3x 1 2 2 x2 2 3x 52 2. The products differ by 2. 3. a. 130 2 112 5 18 b. 598 2 580 5 18 c. 11,128 2 11,110 5 18 d. (x 1 3)(x 1 6) 2 x(x 1 9) 5 x2 1 9x 1 18 2 x2 2 9x 5 18 4. The products differ by 18. 5. The products differ by 32. 6. a. (x 1 2)(x 1 4) 2 x(x 1 6) 5 x2 1 6x 1 8 2 x2 2 6x 58 b. (x 1 5)(x 1 10) 2 x(x 1 15) 5 x2 1 15x 1 50 2 x2 2 15x 5 50 c. (x 1 6)(x 1 2(6)) 2 x(x 1 3(6)) 5 x2 1 3(6x) 1 2(62) 2 x2 2 3(6x) 5 2(62) 5 72 d. (x 1 k)(x 1 2k) 2 x(x 1 3k) 5 x2 1 3kx 1 2k2 2 x2 1 3kx 5 2k2 7. If the numbers increase by any real number k, then the difference of the product is 2k2. 8. 9 8. 9. 10. 11. 12. 13. Enrichment Activity 2-5: Investigating Ratios and Growth Rate in Leaves Students answers will all differ as they have different size leaves. If the length to width ratios are very similar, students should conclude that the rate of growth in their tree or bush is constant. If the length to width ratios vary a lot, they should conclude that the growth rate for their tree or bush is not constant. Students should be assessed on the following characteristics: a. the accuracy of their measurement b. the construction of their data table and scatter plot c. the accuracy of their computations with the calculator d. their knowledge of ratio and average e. the neatness of their work f. their ability to follow directions g. how well they work with others if the activity is done as a group h. their ability to reach a conclusion Enrichment Activity 1-6: Factoring the Sum and Difference of Two Cubes 1. 2. 3. 4. 5. 6. a3 2 b3 5 a3 2 a2b 1 a2b 2 ab2 1 ab2 2 b3 5 a2(a 2 b) 1 ab(a 2 b) 1 b2(a 2 b) 5 (a 2 b)(a2 1 ab 1 b2) (2x 1 y)(4x2 2 2xy 1 y2) (x 2 2y)(x2 1 2xy 1 4y2) (5 2 3d)(25 1 15d 1 9d2) (4x 1 3y)(16x2 2 12xy 1 9y2) a4 2 b4 5 a4 2 a3b 1 a3b 2 a2b2 1 a2b2 2 ab3 1 ab3 2 b4 5 a3(a 2 b) 1 a2b(a 2 b) 1 ab2 (a 2 b) 1 b3(a 2 b) 5 (a 2 b)(a3 1 a2b 1 ab2 1 3) 5 (a 2 b) fa2(a 1 b) 1 b2(a 1 b)g 5 (a 2 b)(a 1 b)(a2 1 b2) a4 2 b4 5 (a2 2 b2)(a2 1 b2) 5 (a 2 b)(a 1 b)(a2 1 b2) (x 2 2)(x2 1 2x 1 4) (x 1 4)(x2 2 4x 1 16) (x 2 4)(x2 1 4x 1 16) (x 1 5)(x2 2 5x 1 25) (x 2 2y)(x2 1 2xy 1 4y2) a3 1 b3 5 a3 1 a2b 2 a2b 2 ab2 1 ab2 1 b3 5 a2(a 1 b) 2 ab(a 1 b) 1 b2(a 1 b) 5 (a 1 b)(a2 2 ab 1 b2) Enrichment Activity 3-2: A Square-Root Algorithm 1. 57 5. 2.6 246 2. 72 6. 4.1 3. 91 7. 5.9 4. 39 8. 9.4 14580AKEA.pgs 3/26/09 12:07 PM Page 247 Enrichment Activity 3-5: A Radical Sequence Enrichment Activity 5-6A: Complex Number Operations,Vectors, and Transformations 1 2 5 3 12 !5 1. 1 12 !5 1 1 5 1 1 !5 2 1. a. 2. 4 1 22 !5 5 2 1 !5 yi B 3. 7 1 23 !5, 11 12 5 !5 4. The sequence has a common ratio, r. A, E 5. 1 12 !5 2 1 5 5 6 1 22 !5 5 3 12 !5 6. Q 1 12 !5 R 5 1 1 2 !5 2 x O 1 5 7. Q 3 12 !5 RQ 1 12 !5 R 5 3 1 4 !5 4 C 5 8 1 44 !5 5 2 1 !5 D 7 1 3 !5 5 8. A2 1 !5B Q 1 12 !5 R 5 1 1 3 !5 2 1 2 5 2 B 5 23 1 5i C 5 25 2 3i D 5 3 2 5i E 5 5 1 3i 5 A b. F 5 23 1 5i 5 B c. Answers will vary: multiplication by i is equivalent to a counterclockwise rotation of 90° about the origin. Multiplication by i2 (or 21) is equivalent to a rotation of 180°. Multiplication by i3 (or 2i) is equivalent to a counterclockwise rotation of 270° about the origin. Multiplication by i4 (or 1) is the identity transformation. d. Answers will vary: Point symmetry in the origin Rotational symmetry of 90° (as well as 180° and 270°) about the origin 11 1 5 !5 9. Q 7 1 23 !5 RQ 1 12 !5 R 5 74 1 104!5 1 15 4 5 2 10. 1.618, the golden ratio Enrichment Activity 4-5: The Method of Finite Differences 1. 2. 3. 4. f(x) 5 x2 2 3x 1 8 f(x) 5 2x2 1 5x 2 3 f(x) 5 3x2 2 4x 2 15 f(x) 5 x3 2 2x2 1 5x 2 3 Enrichment Activity 4-7: The Difference Quotient g 1. a. 2x 1 h b. 2x 2. a. x3 1 3hx2 1 3h2x 1 h3 b. 3x2 1 3hx 1 h2 c. 3x2 3. a. x4 1 4hx3 1 6h2x2 1 4h3x 1 h4 b. 4x3 1 6hx2 1 4h2x 1 h3 c. 4x3 4. a. x5 1 5hx4 1 10h2x3 1 10h3x2 1 5h4x 1 h5 b. 5x4 1 10hx3 1 10h2x2 1 5h3x 1 h4 c. 5x4 5. a. 2x 3x2 4x3 5x4 b. Possible answer: The value of the difference quotient when h 5 0 for f(x) 5 xn is nxn21. 6. a. 6x5 b. 9x8 c. nxn21 g Line symmetry through AOC, through BOC 2. a–c. H yi 30 G 20 10 F A 210 O E D 247 B C 10 x 14580AKEA.pgs 3/26/09 b. B 5 4 2 2i D 5 24 2 8i F 5 216 1 8i H 5 16 1 32i 3. a–b. 12:07 PM Page 248 5. C 5 2 2 6i E 5 212 2 4i G 5 28 1 24i yi A D C O yi x B I a–b. B 5 2 2 3i, C 5 4, D 5 13 c. C 5 (2 1 3i) 1 (2 2 3i) 5 4 D 5 (2 1 3i)(2 2 3i) 5 22 1 32 5 13 d. Both C and D. C 5 (a 1 bi) 1 (a 2 bi) 5 2a D 5 (a 1 bi)(a 2 bi) 5 a2 1 b2 e. P 1 Q 5 PQ 2a 5 a2 1 b2 0 5 a2 2 2a 1 b2 a 5 1 6 "1 2 b2 Since a and b are real numbers, 1 2 b2 $ 0 or 1 $ b2. Thus, P 1 Q 5 PQ when (a, b) 5 (1, 1) or (1, 21). B C A D H x O E Enrichment Activity 5-6B: Quaternions F 1. a. 4 1 6i 1 2j 1 3k; 4(1, 0, 0, 0) 1 6(0, 1, 0, 0) 1 2(0, 0, 1, 0) 1 3(0, 0, 0, 1) b. 27 1 5j 1 8k; 27(1, 0, 0, 0) 1 5(0, 0, 1, 0) 1 8(0, 0, 0, 1) c. 6j 1 9k; 6(0, 0, 1, 0) 1 9(0, 0, 0, 1) d. 23; 23(1, 0, 0, 0) 2. a. (8, 0, 2, 3) b. (0, 4, 7, 21) c. (8, 0, 1, 0) d. (0, 26, 0, 2) 3. a. (16, 4, 6, 6) b. (27, 4, 12, 7) c. (1, 0, 0, 0) d. (8, 0, 7, 9) 4. a. 26j b. 210i c. 24k d. 221 e. 21 f. 2k g. k h. j 5. 263 1 37i 1 27j 1 9k 6. 2a, a real number 8. a. 6 2 7i 2 2j 1 k; 90 b. 3 2 5j 2 2k; 38 c. 28i 2 3j; 73 d. 25i 1 9j 2 4k; 122 8. 6 solutions; 6i, 6j, 6k G B 5 2i C 5 22 1 2i D 5 24 E 5 24 2 4i F 5 28i G 5 8 2 8i H 5 16 I 5 16 1 16i c. (1) H, P, X (2) D, L, T (3) B, F, J, N, R, V, Z 4. Case 1: Multiplication by 0 1 0i is not a transformation of the plane because every point maps to 0, a single point. Case 2: If a 0 but b 5 0, then multiplication by a, is a dilation of a. A dilation of a is a special case of spiral similarity where no rotation occurs. Case 3: If a 5 0 but b 0, then multiplication by bi, is a composition, in either order, of a dilation of b and a counterclockwise rotation 90°. Again, this is a special case of a spiral similarity, but a limiting one. (For example, see the result of multiplying by i in Exercise 1). Case 4: If a 0 and b 0, the transformation is the sum of the two images shown in Cases 2 and 3. This is a true spiral similarity. Enrichment Activity 6-2: Arithmetic Sequences A true spiral similarity occurs when a point is multiplied by a 1 bi where a 0 and b 0. 1–10. Answers will vary. 11. Answers will vary. 12. Possible answer: A 5 52, 4, 5, 76 B 5 51, 3, 6, 86 248 14580AKEA.pgs 3/26/09 12:07 PM Page 249 Enrichment Activity 7-3: Factoring Expressions with Rational and Negative Exponents 1 2. x3 (1 2 x) 3 4. x x2 1 6. 4 7. x 1 15 1 x x2 9. 2(x 2 3) x5 1 11. (x2 1 3) 2 1 1 13. (x4 1 5)(x5 2 1) 1 1 15. (4b2 2 1)(2b2 1 1) 7. 24° 11. 1.179 1 (3) 1 min or 60 hr b. (1) p6 radians/hr (2) 2p radians/hr (3) 120p radians/hr c. (1) p in./hr (2) 16p in./hr (3) 1,200p in./hr 2. a. (1) 2p radians/day p (2) 12 radians/hr 5(x2 2 2) x 1 6. 47° 10. 34° 1. a. (1) 12 hr (2) 60 min or 1 hr 2 1 w5 w12 2 8. 1 2 3c9 1 c c5 10. 5. 42° 9. 43° Enrichment Activity 10-1: Angular Speed and Linear Speed 1 1. y2 (y2 1 1) 3. 1 13 x x5 2 5. 1 2b b 3 4. 1.654 8. 41° 1 12. (x5 1 5)(x5 2 1) 1 1 14. (2y7 1 3)(y7 2 1) 1 1 16. (5x6 2 3)(3x6 2 1) Enrichment Activity 8-5: Finding e b. (1) 12,800p km/day (2) 1,700 km/hr c. 0 km/any time unit. There is no rotation on the North Pole because it lies on the axis of rotation. p 3. a. 3 radians/hr 2. 2.718281823 3. Yes 4. Answers will vary. Enrichment Activity 8-6: State Population Growth Answers will vary by state and with reference used. Sample answers are shown for New York. Population in 1960: 16,782,304 Population in 2000: 18,976,457 1. +2,194,153 2. About 13.1% 3. About 54,854 4. y 5 54,854x 1 16,782,304 where x is the number of years since 1960 5. a. 19,525,004 b. 20,347,814 c. 24,461,864 6. a. 18,976,457 5 16,782,304e40r b. About 0.31% 7. a. 19,595,991 b. 20,528,722 c. 25,902,125 8. Possible answer: The exponential model predicts larger populations than the linear model. 9. About 224 years 10. About 6.9% b. 2,400p km/hr (use r 5 6,400 km 1 800 km) 4. a. 240p radians/min b. (1) 3,360p in./min (2) 280p ft/min c. Yes, 280p ft/min 879.6 ft/min d. 10 mi/hr 5. a. (1) 480p rad/min (2) 8p rad/sec b. (1) 1,400p in./min (2) 24p in./sec 6. a. 24p rad/min b. 360p ft/min c. 1,131 ft Enrichment Activity 10-2: The Angle Between Two Lines Enrichment Activity 9-7: Reflection and Refraction 1. a. 98 or 1.125 2. a. 14 5 or 2.8 b. 48° c. 4p 15 b. 70° c. 7p 8 3. a. 1 b. 45° c. p4 4. a. 31 or 0.3 b. 18° p c. 10 7 9 b. 38° c. 19p 90 5. a. or 0.7 6. a. 82° 1. 35° 2. 17° 3. a. 123,917 mi/s b. 127,572 mi/s c. 139,535 mi/s 249 b. 98° 14580AKEA.pgs 3/26/09 12:07 PM Page 250 Enrichment Activity 11-3: Graphing Combined Functions x 0 p 6 p 4 p 3 p 2 2p 3 3p 4 5p 6 p sin x 1 cos x 1 1.37 1.42 1.37 1 0.37 0 20.37 21 sin x 2 cos x 21 20.37 0 0.37 1 1.37 1.42 1.37 1 x 7p 6 5p 4 4p 3 3p 2 5p 3 7p 4 4p 6 2p sin x 1 cos x 21.37 21.42 21.37 21 20.37 0 0.37 1 sin x 2 cos x 0.37 0 20.37 21 21.37 21.42 21.37 1 1. See above 2. y 1 x p 6 p 3 p 2 2p 3 5p 6 p 7p 6 4p 3 3p 2 5p 3 11p 6 p 7p 6 4p 3 3p 2 5p 3 11p 6 2p –1 b. 21.42; 5p 4 3. a. 1.42; p4 4. See above 5. y c. 2p 1 x p 6 p 3 p 2 2p 3 5p 6 2p –1 6. a. 1.42; 3p 4 7. p 3p 2, 2 , b. 21.42; 7p 4 10. a. c. 2p 2p 8. b. sin x and sin (x 1 p) have opposite values that add to 0 at all values of x. c. Graph 9. a. Max value 1.3 at 1.047 radians b. Min value 21.3 at 5.236 radians c. 2p 250 14580AKEA.pgs 3/26/09 12:07 PM Page 251 Enrichment Activity 11-4: Polar Coordinates 1. Q 32, 3 !3 2 R Part II 6. 2. (0, 2) 3. A22!2, 2 !2B 1 4. Q !3 2 , 22 R 5. (0, 21) 6. (5, 0) 11. 13. A 3 !2, A 5, p 2B p 4B 10. A 4 !2, r 5 2(1 1 sin u) 7p 4 B r 5 2(1 2 sin u) y y 12. (4, p) A 6, p3 B 14. A 4, 5p 6 B x x 4 16. r 5 2 cos u 1 3 sin u 15. r 5 a r 5 2(1 1 cos u) Enrichment Activity 11-8: Graphing Polar Equations Part I 1. x x 3 8. Q23 !3 2 , 22 R 7. (0, 22) 9. y y 7. Each graph is a cardiod (heart) of the same size. Sine graphs are up or down with respect to the y-axis and cosine graphs are right or left with respect to the y-axis. 8. Sine graphs are symmetric with respect to the y-axis. Cosine graphs are symmetric with respect to the x-axis. 9. Enlarging a enlarges the size of the cardiod. Part III 10. y y y y x x r 5 2 sin 2u r 5 2 sin 3u r 5 2(1 2 cos u) y y x x x x r 5 0.2u r 5 2 cos 5u r 5 20.2u r 5 2 cos 4u 11. When a . 0, the spiral opens right; when a , 0, the spiral opens left. 2. Enlarging a enlarges the size of the petals. If b is odd, the graph has b petals. If b is even, the graph has 2b petals. 3. See the answer to Exercise 2. 4. For b odd, graphs involving the sine are symmetric with respect to the y-axis and graphs involving the cosine are symmetric with respect to the x-axis. For b even, both sine and cosine graphs are symmetric with respect to the x-axis, the y-axis, the origin. 5. See the answer to Exercise 2. Enrichment Activity 12-8: Forming Identities 1. 5. 9. 13. f b k g 2. 6. 10. 14. a l c d 3. 7. 11. 15. m i h j 4. e 8. n 12. o Bonus: 251 1. 2 !3 3 2. !3 2 3. 2 4. 212 5. !3 6. !3 3 7. !3 4 8. !3 14580AKEA.pgs 9. 2 !3 2 13. 12 3/26/09 12:07 PM Page 252 10. 1 11. 43 14. 16 3 15. 4 !3 5. s 2 b 5 a 1 2b 1 c 2 b 5 a 1 b 22 2b 1 c 12. 12 5 c 1 a2 2 b 6. s 2 a 5 a 1 2b 1 c 2 a 5 a 2 2a 21 b 1 c Enrichment Activity 13-4: Solving Trigonometric Inequalities 3p 1. p4 # x , p2 or 5p 4 #x , 2 2. p2 , x , 3p 2 5 c 2 a2 1 b 2 c 1 a 2 b c 2 a 1 b ? ? 7. 1 2 cos C 5 ab 2 2 2 5 ab (s 2 b)(s 2 a) 3. p2 # x # 3p 2 4. p6 , x , p or 7p 6 , x , 2p 5. a. p4 , 5p 4 b. Graph y 5 sin x and y 5 cos x and identify intervals where the graph of sin x is below the graph of cos x. The solution is 0 # x , p4 or 5p 4 , x , 2p. 8. (1 1 cos C)(1 2 cos C) 2 5 A ab B s(s 2 a)(s 2 b)(s 2 c) 2 2 1 2 cos2 C 5 A ab B s(s 2 a)(s 2 b)(s 2 c) 2 2 sin2 C 5 A ab B s(s 2 a)(s 2 b)(s 2 c) 2 2 sin C 5 ab !s(s 2 a)(s 2 b)(s 2 c) (Reject negative root since the sine of any angle of any triangle is always positive.) c. p4 # x # 5p 4 6. 7. 8. # x # 7p 4 7p 11p , x , 5p 6 or 6 , x , 6 0 , x # p4 or p2 , x # 3p 4 or p 7p or 3p , x # 2 4 p 4 p 6 9. Area 5 12ab sin C 2 5 12ab ? ab !s(s 2 a)(s 2 b)(s 2 c) , x # 5p 4 5 !s(s 2 a)(s 2 b)(s 2 c) Enrichment Activity 14-4: Heron’s Formula 2 2 10. a. Area 5 12 (5)(12) 5 30 b. Area 5 !15(15 2 5)(15 2 12)(15 2 13) 2 a 1 b 2 c 1. 1 1 cos C 5 2ab 2ab 1 2ab 2 1 b2 2 c2 5 a 1 2ab2ab 5 5 5 5 !15(10)(3)(2) 5 !900 5 30 11. a. Area 5 12 (12)(21) 5 126 (a 1 b) 2 2 c2 2ab f (a 1 b) 1 cg ? f (a 1 b) 2 cg 2ab 2 a 1 b 1 c a 1 b 2 c ? ? 2 2 ab b. Area 5 !27(14)(7)(6) 5 !15,876 5 126 12. a. Area 5 12 (4) Q 4 !3 2 R 5 4 !3 b. Area 5 12 (4)(4) sin 608 5 4!3 2. s 2 c 5 a 1 2b 1 c 2 c 5 a 1 b 12 c 2 2c c. Area 5 !6(2)(2)(2) 5 !48 5 4!3 13. a. 5 a 1 2b 2 c 2 a 1 b 1 c a 1 b 2 c ? ? 3. 1 1 cos C 5 ab 2 2 5 60!3 m2 b. Area 5 !30(30 2 12)(30 2 20)(30 2 28) 2 5 ab s(s 2 c) 4. 1 2 cos C 5 5 5 5 5 /C 5 1208 Area 5 12 (12)(20) sin 1208 5 !30(18)(10)(2) 5 !10,800 2ab a2 1 b2 2 c2 2ab 2 2ab c2 2 (a2 2 2ab 1 b2) 2ab c2 2 (a 2 b) 2 2ab fc 1 (a 2 b)g ? fc 2 (a 2 b)g 2ab 2 c 1 a 2 b c 2 a 1 b ? 2 2 ab ? 5 60 !3 m2 Enrichment Activity 14-7: The Law of Tangents 3 5 1. 210 252 tan 12 (A 2 B) tan 12 (A 1 B) 14580AKEA.pgs 3/26/09 12:07 PM Page 253 2. 100° 3. tan 1 2 (A 2 B) 5 3 210 6. 0.9213 7. The value from step 6 is very close to the calculator value of 0.9215. 8. 2 x 5 12.25, 2 y 5 18.25 sx 5 10.4983, sy 5 15.6365 tan 508 < 20.358 4. A 2 B 5 239.3958 5. A 1 B 5 100, A 2 B 5 39.395 A 5 69.7°, B 5 30.3° 6. c sin 808 26 sin 69.78 26 sin 808 sin 69.78 5 c5 /A 5 69.78 /B 5 30.38 /C 5 808 8. Find c using the Law of Cosines. Then use the Law of Sines to find A or B. 9. a. 90° b. 1 7 c. tan 12 (A 2 B) 5 17 xi 2 2 x sx yi 2 2 y sy 0 4 21.167 20.911 1.063 2 7 20.976 20.719 0.702 5 16 20.691 20.144 0.099 9 14 20.310 20.272 0.084 12 8 20.024 20.656 0.016 17 25 0.452 0.432 0.195 23 19 1.024 0.048 0.049 30 53 1.690 2.222 3.757 r 5 0.8518 A 2 B 5 44.8 Enrichment Activity 16-3: Chi-Square (x2) Test for Goodness of Fit d. /A 5 67.48 /B 5 22.68 6 c sin 908 5 sin 67.48 Problem 1 1. 6 c 5 sin 67.48 c 5 6.5 e. 62 1 2.52 5 42.25 !42.25 5 6.5 6 f. tan A 5 2.5 6 /A 5 tan21 A 2.5 B 5 67.48 10. A 5 36.6° B 5 23.4° c 5 30.5 1 2. 12.75 Amount Expected Frequency (Observed 2 Expected) 2 Expected $0.50 200 2.88 $1.00 150 0.96 $2.00 100 3.24 $3.00 50 5.78 tan B 5 2.5 6 /B 5 tan21 A 2.5 6 B 5 22.68 2. 12.86 3. Yes; 12.86 . 7.81 Problem 2 1. Enrichment Activity 15-8: Calculating the Correlation Coefficient 1. 3.5 5. xi xi 2 2 x yi 2 2 y sx R Q sy R yi c 5 27.3 7. a 5 26 b 5 14 c 5 27.3 Q xi 3. 2.646 Expected Observed (Observed 2 Expected) 2 Expected Face Frequency Frequency 4. 9.639 Q xi 2 2 x yi 2 2 y sx R Q sy R 1 100 2 0.955 3 20.700 0.397 0.752 0.142 1.270 yi xi 2 2 x sx yi 2 2 y sy 3 20.945 21.011 2 6 20.567 4 20 0.189 7 22 1.323 0.960 Total: 2.764 253 113 1.69 100 88 1.44 100 103 0.09 4 100 117 2.89 5 100 95 0.25 6 100 84 2.56 14580AKEA.pgs 3/26/09 12:07 PM Page 254 2. 14.92 3. Yes; 14.92 is greater than the critical value of 11.07. There is sufficient evidence to reject the claim that the die is fair. Problem 3 Answers will vary. 4. a. (1) Students can assign any four digits to success. Example: let the digits 0, 1, 2, 3 represent success. (2) Students execute the randInt(0, 9) command until a success is found. They record the number of executions including the success and record their results. This is repeated a total of ten times. (3) The empirical probability will vary. However, the probability is found by counting the number of trials where success occurred on the fourth execution and dividing by the total number of trials (10). b. Answers will vary. The theoretical probability is .0864. Enrichment Activity 16-4: Geometric Probability Distribution 1. a. b. 2. a. b. 3. a. b. .043 .037 5 36 .076 .006 .004 254 14580AKET.pgs 3/26/09 12:10 PM Page 255 Answers for Extended Tasks Chapter 1 6. Draw the squares of each walking distance for each point. Points that are the same walking distance from (8, 9) and (6, 5) will be those located on squares of the same size relative to both points. Going for a Walk 1. 10 units for each route. Routes will vary. Examples: (1, 1) to (7, 1) to (7, 5); (1, 1) to (1, 5) to (7, 5); (1, 1) to (1, 3) to (7, 3) to (7, 5) y 2. a. 15 units b. 19 units c. 21 units same distance 3. a. (0, 6), (1, 5), (2, 4), (3, 3), (4, 2), (5, 1), (6, 0), (7, 1), (8, 2), (9, 3), (10, 4), (11, 5), (12, 6), (11, 7), (10, 8), (9, 9), (8, 10), (7, 11), (6, 12), (5, 11), (4, 10), (3, 9), (2, 8), (1, 7) y O 1 2 3 4 5 same distance 5 4 3 2 1 (6, 5) (6, 12) O (0, 6) (8, 9) (6, 0) 7. Draw the line segment connecting (8, 9) and (6, 5). The midpoint of the line segment is (7, 7). The line perpendicular to the segment through (7, 7) is the perpendicular bisector and all points on the perpendicular bisector of a segment are equidistant from the endpoints of the segment. 8. a. 7 2 2 or 2 2 7 5 5 units b. 5 2 (24) or 24 2 5 5 9 units c. x2 2 x1 or x1 2 x2 9. a. 20 2 5 or 5 2 20 5 15 units b. 28 2 15 or 15 2 (28) 5 23 units c. y2 2 y1 or y1 2 y2 10. a. 10 2 2 1 10 2 4 5 14 units b. 211 2 (25) 1 7 2 1 5 22 units c. x2 2 x1 1 y2 2 y1 (12, 6) (6, 6) x x b. A square c. The set of all points outside the square d. The set of all points inside the square 4. a. !72 b. 12 units c. The walking distance 5. a. When the points are on the same horizontal or vertical line b. Never Chapter 2 Electronic Technician:Applying Rational Equations in the Workplace a. A series circuit has the resistors positioned to provide a single path for current flow. A parallel circuit has the resistors positioned to provide two or more paths for current flow. 255 14580AKET.pgs 3/26/09 b. 12:10 PM Page 256 and BC. Since AB 5 1 and BC 5 3, DB is the mean proportional between 1 and 3, or 1 : x 5 x : 3. Since the product of the means equals the product of the extremes in any proportion, x2 5 3 or x 5 !3. Exercise II 1. R1 R2 R3 c. D R2 R1 √7 R3 M 7 A 1B C d. Series Circuit: RT 5 R1 1 R2 1 R3 1 c 1 Rn The total resistance in a series circuit is the sum of the individual resistances in the circuit. Parallel Circuit: 1 5 1 1 1 1 1 1 c 1 1 RT R1 R2 R3 Rn The reciprocal of the total resistance in a parallel circuit is the sum of the reciprocals of the individual resistances in the circuit. e. 1. (1) Series circuit (2) R3 5 7,000 ohms 2. (1) Parallel (2) R2 5 30 ohms 3. (1) Parallel circuit (2) R4 5 6,000 ohms 4. (1) Combination series-parallel circuit (2) R6 5 40 ohms D √5 A1 B DB 210 16 in. for !7 4 3. !5 : !7 5 216 4 210 16 Finding Square Roots Geometrically Exercise I 5 94 4 21 8 D 4. !35 7 B C 4 2. DB 216 in. for !5 Chapter 3 A M 5 5. !35 7 0.8452 C M 5 67 < 0.8571 6. 0.012 Chapter 4 The Inverse Variation Hyperbola 1. 2. 3. 4. 5. Activity 1 Students should discover that the product of the force required to balance the weight and the distance from the fulcrum is constant and equal to the weight placed on the left side peg. 2 a. 4 in. b. 3 in. See construction above. See construction above. See construction above. Yes. Since DB is the altitude to the hypotenuse of ACD, it is the mean proportional between AB 256 14580AKET.pgs 3/26/09 12:10 PM Page 257 j Activity 2 The same relationship should exist. Activity 3 Verbal Description: The product of the force, f, exerted and the distance, d, of the spring from the fulcrum is constant and equal to the weight, w, on the left side of the number balance. Algebraic Description: w 5 fxd or xy 5 c Activity 4 Graphs will differ, but should be a hyperbola (one branch) in Quadrant I. Activity 5 The curves will be the other branch of the hyperbola drawn in Activity 4. This branch will be in Quadrant III. The curves will have the same equations as stated in Activity 3. Inverse variation is when two quantities change or vary such that their product is a nonzero constant. That is, xy 5 k or y 5 kx, x not equal to zero. 6. S2j . 1 1 2 . The series does not have a limiting sum. Since the partial sums of the harmonic series have been j shown to be greater than 1 1 2 which is unbounded, the partial sums do not approach a limit. 7. 11 8. 31 Chapter 7 Holes, Holes, and More Holes: An Exponential Investigation Part I Task 1 The Harmonic Series 1. n1 approaches 0. 1 S4 5 25 12 5 22 S2 5 32 17 S5 5 137 60 5 260 5 S3 5 11 6 5 16 27 S6 5 49 20 5 260 0 1 2 3 4 5 # of holes 1 2 4 8 16 32 # of holes expressed as a power of 2 20 21 22 23 24 25 a. Answers will vary, but should be something like: “The total number of holes doubles with each fold.” or “The number of holes is a power of 2, the power being the number of folds.” b. 2n c. h 5 2n Chapter 6 2. S1 5 1 # of folds Task 2 3. Answers will vary. Some students will think the series has a sum because the nth term approaches 0. Others may see that the sums can be made as large as is required. The series does not have a sum. 4. S23 5 S8 5 1 1 12 1 13 1 14 1 c 1 18 # of folds 0 1 2 3 4 5 # of holes 2 4 8 16 32 64 # of holes expressed as a power of 2 21 22 23 24 25 26 a. Answers will vary, but should be something like: “The pattern is similar, but it begins with 21 rather than 20.” b. 2 3 20, 2 3 21, 2 3 22, 2 3 23, 2 3 24, 2 3 25 c. h 5 2(2n) 5 S4 1 15 1 16 1 17 1 18 Since S4 . 1 1 22 : . 1 1 22 1 15 1 16 1 17 1 18 . 1 1 22 1 18 1 18 1 18 1 18 511 5. S24 5 S16 5 A1 1 1 2 Task 3 3 2 1 1 c 1 18 B 1 A 19 1 c 1 16 B 1 . 1 1 32 1 19 1 c 1 16 .1131 1 1c1 1 .11 16 0 1 2 3 4 5 # of holes 3 6 12 24 48 96 # of holes expressed 3320 3321 3322 3323 3324 3325 as a power of 2 Since S8 . 1 1 32 : 2 4 2 # of folds h 5 3(2n) 16 257 14580AKET.pgs 3/26/09 12:10 PM Page 258 Part II h 102 h 5 3(2n) 96 90 84 78 72 Number of holes 66 h 5 2(2n) 60 54 48 h 5 2n 42 36 30 24 18 12 6 0 1 2 4 3 5 n Number of folds Answers will vary. Students should observe that the graphs start out very low at about the same point, but rise rapidly. As the constant increases for each graph, the graph rises more sharply than the previous one. Chapter 8 Part III Answers will vary. For example, “Yes. It is appropriate because as n, the number of folds, increases, h, the number of punched holes, increases rapidly.” or “. . . it rises a lot sharper than a quadratic.” or “. . . it increases exponentially.” Calculating the Magnitude of an Earthquake: A Mathematical Application 1. 3 2. 2.5 3. 5 4. 4.1 5. 4 Part IV Answers will vary. 258 14580AKET.pgs 3/26/09 12:10 PM Page 259 Chapter 9 Trigonometry in Aviation 1. cloud Cloud height Observer's eye u 70° m Parabolic light source n 1,000 ft Ground Let the segments of the base of the triangle be represented by m and n. n Then: cot u 5 m h and cot 70° 5 h , or m 5 h cot u and n 5 h cot 708 Since m 1 n 5 1,000 we will add these 2 equations getting m 1 n 5 h cot u 1 h cot 708 1,000 5 h(cot u 1 cot 708) 1,000 h 5 cot u 1 cot 708 2. a. 400 ft b. 730 ft c. 2,300 ft 3. 86° 9–10. Answers will vary according to data chosen by student. Chapter 12 4. 58° 1. Chapter 11 Temperature Temperature 1. 47° 3–5. 70 60 50 40 30 20 10 0 By Formula 2. 47.5° 0 1 2 3 4 5 6 7 8 9 10 11 12 Month 6. Sine curve 7. Approximately 365 days 8. 23.5 259 From Calculator Angle Sin Cos 1° 0.017460 0.999850 0.0175 Sin 0.9998 Cos 2° 0.0349 0.9994 0.0349 0.9994 3° 0.0524 0.9986 0.0523 0.9986 4° 0.0698 0.9976 0.0698 0.9976 5° 0.0872 0.9962 0.0872 0.9962 6° 0.1046 0.9945 0.1045 0.9945 7° 0.1219 0.9926 0.1219 0.9925 8° 0.1392 0.9903 0.1392 0.9903 9° 0.1565 0.9877 0.1564 0.9877 10° 0.1737 0.9848 0.1736 0.9848 14580AKET.pgs 3/26/09 12:10 PM Page 260 Chapter 14 2. Possible answer: The values were very close. 3. Land for Sale: A Trigonometric Investigation 1. 45,400 ft2 2. 1 acre 3. 18 lots 4. $5,500 5. $99,000 6. 52% Angle Tan (by formula) Tan (calculator) 1° 0.0175 0.0175 2° 0.0349 0.0349 3° 0.0524 0.0524 4° 0.0700 0.0699 5° 0.0875 0.0875 Chapter 15 6° 0.1052 0.1051 7° 0.1228 0.1228 Taking a Survey: Designing a Statistical Study Answers will vary. 8° 0.1406 0.1405 9° 0.1585 0.1584 10° 0.1764 0.1763 Possible answer: The values are again very close. Chapter 13 Find the Letter: A Trigonometric Puzzle 1. a. sin2 x b. 135° sin x c. 1 2 cos x e. p 3 d. cot x f. 210° g. csc x h. !5 5 i. 25° and 225° 63 j. 65 k. 7p 12 2–3. I LOVE TRIG. DO YOU? 260 14580AKST.pgs 3/26/09 12:10 PM Page 261 Answers for Suggested Test Items xy Chapter 1 5. 2z (x 2 0, y 2 0, z 2 0) 1. 19 2. 5 3. 4 4. 213 5. a. All integers b. All integers n $ 0 c. { } or 6. x 5 7 7. c 5 24 8. y 5 9 9. x 5 213, 7 10. m 5 24, 1 11. {3, 4, 5, . . . } or k . 2 12. {2, 3, 4, 5, 6, 7, 8, 9} or 2 # x # 9 13. n3 2 7n2 1 n 1 2 14. c2d2 1 5 15. 10x3 2 8x2 1 5x 1 3 16. 22y2 2 2y 1 4 17. 90 min on math, 62 min on science 18. 6 ft and 34 ft 19. Kate is 9, her mother is 45. 20. {2, 3, 4, 5, 6} or 2 # x #6 21. {24, 23, 22, 5, 6, 7, . . . } or x , 21 or x . 4 22. {29, 28, 27, 26, 25, 24, 23, 22, 21, 0, 1, 2, 3, 4, 5, 6} or 29 # x # 6 23. All integers 24. { } or 25. 22x3 2 12x2 2 18x 26. 24y2 2 4y 27. 9x4 2 33x3 1 30x2 28. 22x 1 8 29. 5x3 2 x2 1 16x 1 16 30. a. 2x2 1 3x b. 189 in.2 31. (a 1 b)(5 1 b) 32. (5x 1 6)(2x 2 3) 33. (x 2 3y)(x 1 3y) 34. n(4n 2 1)(n 2 1) 35. 2x3(2x 2 1)(2x 1 1) 36. (x2 1 4)(x 2 3) 37. x 5 26, 5 38. y 5 21, 4 39. x 5 27, 23 40. x 5 28, 0 41. 11 and 12 42. 28 in. 43. x , 3 or x . 4 44. {23, 22, 21, 0, 1} or 23 # x # 1 45. x , 0 or x . 6 46. {24, 23, 22, 21, 0, 1, 2, 3, 4, 5, 6} or 25 , x , 7 6. x (y 2 23) 7. 2(x 1 7) A x 2 23 B y 2 6 8. y 2 1 (y 2 1, y 2 4) 2 9. 3a (a 2 0, a 2 2b) 10. x1 (x 2 0, 1) 11. 2(c 1 1) 3c 1 12. 2a (a 2 0) 13. 3(a 42 3) (a 2 3, 23) 14. 76 (x 2 21) 5 15. x 2 1 (x 2 1, 21) 2y 1 25 16. y2 2 25 (y 2 5, 25) 1 1 17. 3x9x 1 1 A x 2 0, 3 , 23 B 2 1 2 18. x2 5x 2 4x 1 3 ft (x 2 1, 3, 23) 19. 4 20. 5 21. 3 22. 8 23. 55 24. x (x 5 0, 1, 21) 25. 3b a (a 2 0, b 2 0) x 1 2 2 26. 2 (x 2 2, 22) 27. x 2 8 (x 2 0, 22) 28. 31. 32. 33. 35. 12 29. 212, 2 30. 23, 24 5 Express: 60 mph, freight: 40 mph 4 and 6 34. y . 2 0 , b , 3 or b . 6 36. x . 5 2 4x 1 1 Bonus: a. 4x(2x 2 1 1) m2 1 m 1 1 b. m3 1 m2 1 2m 1 1 Bonus I: a2 2 b2 2 c2 1 2bc 5 a2 2 (b2 2 2bc 1 c2) Chapter 3 5 a2 2 (b 2 c) 2 1. 3. 5. 7. 9. 5 fa 1 (b 2 c)gfa 2 (b 2 c)g 5 (a 1 b 2 c)(a 2 b 1 c) Bonus II: x , 25 or 23 , x , 2; the product is negative when an odd number of factors are negative. If x , 25, all three factors are negative. If x is between 23 and 2, one factor is negative. 1. 0.416 2. 3. 4 9 4. Rational Irrational Irrational a # 24 or a $ 10 All real numbers 11. 4 Chapter 2 5 8 (c 2 0, 21, 22) 4 33 9 12. 11 Rational Rational 25 , x , 5 21 # x # 7 21 13. 0.2 14. 6!5 15. 7b !3 17. 18. 7!5 19. 11!6 21. 29!2 22. 1 5x x 6y2 #y 20. 4!3 261 2. 4. 6. 8. 10. 2 16. 2a2 !7 14580AKST.pgs 3/26/09 12:11 PM 23. 27 1 10 !2 25. 9 !2 1 6 27. 2 1 !5 29. Page 262 13. p(x) 5 (x 1 6)(x 1 2)(x 2 3) 5 x3 1 5x2 2 12x 2 36 14. a. x2 1 2x 2 8 b. 2x2 1 4 15. a. 4 b. !10 16. a. (3x 2 1) 2 5 9x2 2 6x 1 1 b. 3x2 2 1 17. a. 3 b. 3 24. 36 26. 2 28. 12 3 2 !5 2 30. 2 !5 1 2 !2 31. 3 2 2 !2 32. 4 !x 1 4y x 2 y2 34. x 5 !7 2 1 36. x 5 1, 2 33. a 5 14 35. b 5 5 18. y 5 12x 1 12 19. y 5 x 2 7 20. y 5 22x 1 10 21. a. I b. IV 22. a. Yes b. No 23. a. (x 1 3)2 1 (y 2 1)2 5 4 b. Center 5 (23,1), radius 5 2 24. (x 1 4)2 1 (y 2 3)2 5 25 25. 12 5 5 2.4 26. a. {3, 22} b. 22 , x , 3 c. x , 22 or x . 3 4 4 4 30 4 10 30 37. !330 5 ! 4 4 5 # 81 5 # 27 "3 4 5 4 15 # 9 5 # 27 4 15 4 10 4 5 # 27 . # 27 , so # 9 . 38. 1.7m, 5.0m, 5.3m ! 30 3 4 39. Length 5, width 4 Bonus: r r √2 r c. 18 c. 21 27. a. b. {1, 4} y 6 5 4 3 2 1 r r r x 22 21 1 2 3 4 5 6 21 22 23 r 2r 1 2r !2 5 16 !2 r 5 1 81!2!2 28. a–b. c. No y a. r 5 8 !2212 16 r 5 16 2 8 !2 b. Chapter 4 1. a. {22, 21, 0, 1, 2} b. {1, 2, 3} c. Yes 2. a. {x : 23 # x # 3} b. {y : 23 # y # 3} c. No; the function is not one-to-one. 3. a. {x : x # 9} b. {y : y $ 0} c. Yes 4. a. {x : 0 # x # 3} b. {y : 23 # y # 3} c. No; the relation is not a function. 5. a. {x : 21 # x # 1} b. {y : 21 # y # 1} c. No; the relation is not a function. 6. a. 5 7. a. 21 b. 65 b. 217 1 8. 24 9. Domain 5 {all real numbers} Range 5 {26} 10. 10 11. a. c 5 3r b. Yes 12. g(x) 5 8(x 2 6) 2 4 5 8x 2 46 x O 29. a. xy 5 6 b. y 6 4 2 2 26 24 22 O 22 24 26 c. $27 Bonus: x 5212 , 2 262 4 6x 14580AKST.pgs 3/26/09 12:11 PM Page 263 Chapter 5 42. x , 0 or x . 6 44. 212 , x , 12 1. Since y 5 2(x 2 3)2 2 12, the graph of y 5 x2 must be stretched vertically by a factor of 2 and translated 3 units right and 12 units down. 2. 24, 10 3. 26, 3 4. 17i 5. 6i !2 6. 227 7. 40 !5 8. 280 9. 21 1 i 10. 1 11. 2i 12. i 13. 10 2 11i 14. 21 2 i 15. 5 2 12i 16. 21 2 2i 17. 5 2 i 18. 7 1 8i 19. 16 2 16i 20. 1 6 3i 21. 25 6 2i 23. a. b. 26. a. b. 24. a. b. 27. a. b. 22. 12 6 12i 25. a. 33 b. (3) 28. a. 24 b. (4) 41 (3) 220 (4) 29. 1 9 (1) 0 (2) 32. x2 1 3x 2 54 5 0 34. 17, 19 36. 12 , 2 Chapter 6 1. a. 30, 36, 42 2. a. 1, 3, 5 3. a. 10, 1112 , 13 4. a. 212, 14, 218 38. 1 6 !3 y (1, 4) y 5 x2 1 2x 1 1 (22, 1) x O 55 31. 127 8 1 2 35. a. a1 5 1.8, r 5 1.5 36. a. a1 5 5, r 5 225 x 3,280 32. 1,310.72 g 34. a. a1 5 6, r 5 (2, 1) O b. (2, 1) 41. a. 26. 24. 1 4 27 27. 18, 36, 72 or 218, 36, 272 28. 2105 29. 513 y 2x 1 y 5 5 1 n21 23. 384 25. 27,168 b. (22, 1) (1, 4) 40. a. y2 b. an 5 28 A 212 B 22. 22712 y5x13 x2 b. an 5 6 1 6(n 2 1) 5 6n b. an 5 27 1 2(n 2 1) 5 2n 2 9 b. an 5 4 1 32 (n 2 1) 5 32n 1 212 64 128 5. a. 32 b. an 5 25 (2) n21 5, 5, 5 6. a. 21, 1, 21 b. an 5 21(21)n21 5 (21)n 7. 89 8. 270 9. 8 10. 17 11. 14 12. 44.51 13. 80, 105 14. 36, 54, 72, 90 15. 6 16. 21 17. a. 2 1 5 1 10 1 17 1 26 1 37 b. 97 18. a. 12 1 17 1 22 1 27 1 32 1 37 b. 147 19. 100 20. 296 21. 2,430 33. x2 2 14x 1 58 5 0 35. 12 1 12!2 < 28.97 ft 37. 1 6 2i 39. a. Bonus I: a. x2 2 2!2x 1 1 5 0 b. 4 c. Rules for the discriminant apply only when a, b, c are rational. In this equation, b is irrational. 4 Bonus II: P 5 20 5 15 since c 5 17, 18, 19, or 20 will 2 make b 2 4ac negative. 31. 32 30. 4 43. 23 # x # 2 30. 27 33. $3,374.59 b. 12 b. No sum b. 25 7 Bonus I: 80 ft 1 1 1 Bonus II: a. 12 1 16 1 12 1 20 1 30 1c b. No, there is no common ratio. c. S1 5 12, S2 5 23, S3 5 34, S4 5 45, S5 5 56 20 d. S10 5 10 11 , S20 5 21 y y 5 x2 2 x O e. 1 f. S5 5 A 1 2 12 B 1 A 12 2 13 B 1 A 13 2 14 B x y5x22 1 A 14 2 15 B 1 A 15 2 16 B 1 g. Sn 5 1 2 n 1 1 b. No real roots (1 1 i, 21 1 i) (1 2 i, 21 2i) 263 14580AKST.pgs 3/26/09 12:11 PM Page 264 3 b 21. N 5 b !a 22. N 5 a2# c c 23. b 2 1 24. 11 25. 57 26. a. (1) 5.0969 (2) 2.1761 (3) 5.0969 (4) 2.1761 b. 3 log 50 5 log 503; log (3 3 50) 5 log 3 1 log 50 27. 1.53 28. 2.26 29. 4.46 30. 20.9566 31. 2.3036 32. 4.2324 33. 36 years 34. $6,922.14 35. 1.407 36. 2.089 Chapter 7 1. 3 2. 70 3. 125 4. 4 5. 21 6. 52 5 2.5 7. 81 5 0.125 8. 729 1 9. 664 10. 4,096 11. x2y2 2 12. 3ab 13. 125c3d4 14. y2 15. 16. 7 1 1 66 y5 18. 4x2y3 17. (2x) 2 4 19. "(3x) 3 x7 4 y3 20. 5x5 !y 21. a–b. 37. log 25 5 log 2 2 log 5 y 5 0.3010 2 0.6990 5 20.398 38. log x y5 (54 ) 1 2x ( 54) x 1 O 21 c. y 5 A 45 B 22. 16 y5 x 5 A 54 B 24. 1, 3 25. 2 26. 3 27. 34 28. $3,249.43 29. $8,976.16 30. $12,523.23 22 20 18 16 14 12 10 8 6 4 2 4. a. 60 5 x b. 1 5. a. 25 5 x b. 125 7. a. 4x 5 32 1 8. a. 3x 5 27 b. 23 Bonus: logx 4 1 logx9 logx (4 ? 9) x2 x 3. y 5 6x 6. a. x4 5 4 b. !2 5 5 5 5 2 2 36 6 Chapter 9 9. a. x–2 5 0.04 10. 5 11. 45 or 0.8 7 1. a. 25 25 e. 24 12. log 0.01 5 22 13. log x 5 log a 1 2 log b 2. 1,141.7 ft b. 5 2 y 5 log5 x c. Reflection in y 5 x Chapter 8 3 2 y 5 5x O 2 4 6 8 10 12 14 16 18 20 22 x Bonus I: 2a 1 Bonus II: They are equal; both equal x6 when converted to exponential form. 2. y 5 2log8 x 5 log 5 2 log 2 5 0.6990 2 0.3010 5 0.398 39. log 52 5 2 log 5 5 2(0.6990) 5 1.398 40. a–b. y x 23. 12 1. y 5 log7 x 5 2 b. 5 16. log x 5 2 log a 1 2 log c 2 6 log b 17. log x 5 23 log a 1 13 log b f. 25 7 7 g. 25 7 h. 24 d. 2!2 e. f. 3 213 h. i. !2 4 j. A 1, !2 4 B 4. a. !5 3 3 5 3 !5 d. !5 5 264 7 d. 24 c. g. 18. log x 5 2 log a 1 log b 1 12 log c 2 !a 19. N 5 abc 20. N 5 (bc) 2 c. 24 7 b. 2 !2 3 3. a. 13 14. log x 5 12 log a 2 log b 2 3 log c 15. log x 5 log c 1 32 log b 2 52 log a b. 24 25 3 5 3 !2 4 2 !2 2 !2 2 3 2 5 2 !5 b. !5 5 e. !5 2 1 2 !2 5 !2 4 c. 32 f. 23 14580AKST.pgs 3/26/09 5. IV 8. 135° 11. !2 2 14. 2!3 17. !3 3 20. 23. 26. 27. 28. 29. 30. 33. 20.0523 20.6000 3.1716 a. 56° a. 67° a. 76° 20.9 2cos 30° 36. 0.18 12:11 PM 6. III 9. 310° Page 265 28. 6p 26. 2p6 29. 6p2 27. 6p2 30. 2p6 31. 24 25 32. 12 5 33. 25 35. 21 36. 35° 25. p6 7. II 10. 180° 2!3 2 12. 21 13. 15. 21 16. 22 18. !2 34. 2 !3 2 19. 0.7071 37. 30° 22. 0.3739 25. 1.2799 Bonus: 16 25 21. 3.8637 24. 1.7013 b. b. b. 31. 34. 55° 499 66° 569 76° 089 225° cos 20° 37. 0.00 2 38. 181 Chapter 11 1. a. 1 c. 32. 2tan 40° 35. tan 67.5° 38. 1 sin u Bonus I: A1, !3B Bonus II: 1 5 63.4°, 2 5 26.6°, 3 5 63.4° 2. a. 3 c. Chapter 10 1. 4. 7. 315° 10. 4.2 in. 13. 1 2 !2 2 16. cos12 u 19. p 4 22. 2p6 2. p 10 3. 24p 3 5. 108° 6. 2540° 8. 80° 9. 2.4 or 12 5 11. 0.4845 12. 1.4506 14. 16 15. !6 17. sin1 u 20. p2 23. p3 1 0.5 20.5 21 39. sin u 5p 12 17p 6 b. 2p 2u 18. cos sin2 u 21. 2p4 3 2 1 y x p p p 2p 5p 7p 4p 3p 5p 11p 6 3 2 3 6 p 6 3 2 3 6 2p b. 4p 3 2 1 21 22 23 24. 2p 3 x p p p 2p 5p 7p 4p 3p 5p 11p 6 3 2 3 6 p 6 3 2 3 6 2p b. p 21 22 23 3. a. 2 c. y y x p p p 2p 5p 7p 4p 3p 5p 11p 6 3 2 3 6 p 6 3 2 3 6 2p 4. a–b. y y 5 2sin x 2p 2 5p 6 2 2p 3 2 p2 2 p3 1 2 p6 c. y 5 cos 2x 265 21 y 5 cos 2x p 6 p 3 p 2 d. y 5 2sin x x 2p 3 5p 6 p e. 3 14580AKST.pgs 3/26/09 12:11 PM 5. a. 1 b. 2p 6. a. 2 b. 2p 7. a. 1 b. p 1 2 p 2 8. a. b. p 2B 9. a. y 5 2 sin A x 1 b. y 5 2 cos x Page 266 c. 2p4 c. p3 c. 2p2 c. p6 or y 5 22 sin A x 2 14. 0.96 or 24 25 15. 0.8 or 45 19. 0.75 or 34 20. 1,210.8 ft ? sec x 21. a. csc x ? tan x 5 sin x ? 1 1 ? 5 sin x cos x cos x b. Ux : x (2`, `) The values are increasing to 21. 15. 260° !3 60° 17. 230° a. y 1 1 cos x 5 cos x 2 np 2 V 2 ? cos x cos2 x csc2 x 5 sin2 x b. 5x : x 2 np6 23. a. 25 210 215 220 b. 20 sec or 13 min c. 23 ft 19. No interval satisfies the condition; since sec x 5 cos1 x , if cos x is increasing, sec x must be decreasing. b. sin 2x 2 sin x ? sin x cos 2x 1 cos x 5 cos x 1 1 2 sin x cos x 2 sin x ? sin x 2 cos2 x 1 cos x 2 1 5 cos x 1 1 sin x(2 cos x 2 1) sin x ? (cos x 1 1)(2 cos x 2 1) 5 cos x 1 1 sin x sin x cos x 1 1 5 cos x 1 1 ✔ 5p U x : x 2 p 1 2pn, p 3 1 2pn, 3 1 2pn V 0.5 2 cos x ? 2 cos x 5 sin2 x sin2 x cos2 x x p 2 p 3p 2 2p 21 b. Ux : x 2 b. p c. Tnp, 0 or Rx5np2 for any integer n. np V 2 2 cos x 2 sin x x 5 2 cos ✔ 2 sin x Bonus: Let AB 5 BC 5 CD 5 DE 5 s s tan y 5 3s 5 13 s tan y 5 2s 5 12 Chapter 12 1. 3. 5. 7. cot u True False a. 219 2. 4. 6. b. 8. 3 4 9. or 0.75 2x 5 cos sin2 x ✔ sec x 2 1 1 sec x 1 1 ? 2 cos x (sec x 1 1)(sec x 2 1) 5 sin2 x ? 2 cos x 2 sec x sec2 x 2 1 5 sin2 x 2 sec x ? 2 cos x tan2 x 5 sin2 x y 20.5 cos2 x sin2 x ? 2 cos x 24. a. sec x1 1 1 1 sec x1 2 1 5 sin2 x 1 p 2 3p 2 2p 2 2 ✔ ? cot2 x 22. a. cos2 x 1 cos2 x ? cot2 x 5 ? cos2 x 2 2 cos x(1 1 cot x) 5 sin2 x x Bonus: a. 17. 20.28 or 27 25 18. 0.936 or 117 125 11. Ux : x 2 p2 1 np V 20 15 10 5 !2 11. !6 2 4 !2 13. 2 !6 1 4 16. !2 10 p 2B 10. a. y 5 2sin 32x b. y 5 32 cos 32 A x 1 p3 B or y 5 232 cos 32 A x 2 p3 B 12. 13. 14. 16. 18. !2 10. !6 2 4 12. 2 !3 2 csc u True True III tan (x 1 y) 5 1 11 3 2 1 2 13 ? 12 51 Since x and y are acute angles, (x 1 y) 5 45°. tan z 5 ss 5 1, so z 5 45°. Finally, x 1 y 1 z 5 45° 1 45° 5 90°. 23 !7 8 266 14580AKST.pgs 3/26/09 12:11 PM Page 267 Chapter 13 1. 3. 5. 7. 9. 11. 13. 15. 17. 19. 21. Chapter 15 {210°, 330°} {45°, 135°, 225°, 315°} {120°, 240°} {0°, 180°} {0°, 45°, 180°, 225°} {90°, 120°, 240°, 270°} 114°, 246° 34°, 82°, 214°, 262° 34°, 180°, 326° 80°, 180°, 280° 7p 11p 0, p6 , 5p 6 , p, 6 , 6 2. 4. 6. 8. 10. 12. 14. 16. 18. 20. {45°, 225°} {30°, 150°, 210°, 330°} {210°, 270°, 330°} {30°, 90°, 150°, 270°} {120°, 240°} {60°, 120°, 240°, 300°} 101°, 259° 52°, 128° 58°, 148°, 238°, 328° 54°, 147°, 213°, 306° 1. Census 2. a. 21 b. 21.5 c. 22 d. 20 e. 22 f. 6 g. 2.83 h. 1.68 3. a. 15 b. 81 4. a. (1) 2 (2) 2 (3) 2 (4) 1.48 b. (1) 2 (2) 1 (3) 0 (4) 2.21 c. For Mrs. Alvarez’s data, mean 5 median 5 mode, 70% are within one standard deviation of the mean, 95% within two standard deviations, and 5% more than two standard deviations. For Mr. Kazin’s data, the mean, median, and mode are unequal and 85% are within one standard deviation of the mean. Mrs. Alvarez’s data more closely resembles a normal distribution. 5. a. 78°F b. 5.54 6. a. 21.5 b. 2.5 7. a. The ACT; z-score for SAT 1.24, z-score for ACT 5 1.48 b. The ACT since her score on the ACT is farther from the mean than on the SAT. 8. a. 34% b. 2.5% 9. 76.1% 10. 6,100 11. Close to 21 12. Close to 0 13. a. y 5 17.676x 1 84.249 sales 5 17.676 (years) 1 84.249 b. $261,000 c. $703,000; extrapolation 14. a. 50 45 40 35 30 25 20 15 10 5 22. 20°, 40°, 70°, 100°, 140°, 160° Bonus: sin x 5 cos x tan x 5 1 tan x 5 61 6 x 5 458, 1358, 2258, 3158 Chapter 14 3 !2 1. Q 3 !2 2. A26 !3, 6B 2 , 2 R 3. A0.375, 20.375!3B 4. a. 10 b. 37° 5. 30° 6. 16 7. 126° 8. a 5 12.1, c 5 17.9 9. 135.9 sq units 10. 661 cm2 11. 162.1 nautical miles 12. 29 ft 13. 39.2 in. 14. 47° 15. a. 13° b. 480 ft 16. a. 2 b. B 5 53.5°, C 5 86.5°, c 5 12.4 or B 5 126.5°, C 5 13.5°, c 5 2.9 Bonus: First find side b using known values: sin B 5 sinc C S b 5 csin C (We know C because C 5 180° 2 A 2 B.) Therefore: Area 5 12bc sin A Stores b sin B sin B 5 12 A csin C B c sin A c2 sin B sin A 5 2 sin (180 2 A 2 B) 2 sin A sin B 5 2csin (A 1 B) 1 2 3 4 5 6 Year b. Exponential; y 5 0.533(1.928)x c. 196 stores 267 7 14580AKST.pgs 3/26/09 12:11 PM Page 268 Chapter 16 1. 720 4. 1 7. 27,720 2. 210 5. 120 8. 58,464 10. 12! 4! or 19,958,400 21. 22. 23. 24. 25. 26. 3. 190 6. 1099 9. 54 10! 2! 12! 4! 11. 7! 12. 4!3! 5 35 3 1 5 124 3 3 11 5 11 14. !3 4 20 R10120 1 c 1 Q 20 5 220 or 1,048,576 Bonus: 5 people are needed Use 1 2 P(exactly 0) If n 5 4, 1 2 4C0(.1) 0 (.9) 4 5 1 2 .6561 5 .3439 (less than 40%) If n 5 5, 1 2 5C0(.1) 0 (.9) 5 5 1 2 .59049 5 .40951 (greater than 40%) 5 15. Q 28 RQ 10 3 RQ 1 R 5 16,800 5C4 ? 15C1 1 5 C5 20 C5 17. 1 2 4C0 A 13 B 0 A 23 B A 13 B 4 18. 10C4 19. 20C13 20. 25C0 6 A 14 B A 34 B A 109 B 13 25 < .53701 1 5 204 A 23 B 4 65 5 1 2 16 81 5 81 1,120 5 19,683 < .0569 7 < .0002 1 1 25C1 A 10 B 1 A 109 B 24 1 1 25C2 A 10 B 5t4 27 27. (8n3 2 36n2 1 54n 2 27) in.3 20 19 1 20 18 2 20 0 28. (1 1 1) 20 5 Q 20 0 R1 1 1 Q 1 R1 1 1 Q 2 R1 1 13. (.3)(.6) 1 (.3)(.4) 1 (.7)(.4) 5 .58 or 58% 16. .8643 .5561 x6 1 12x5 1 60x4 1 160x3 1 240x2 1 192x 1 64 81x4 1 216x3y 1 216x2y2 1 96xy3 1 16y4 2280a4b3 2 A 109 B 23 268 14580AKSAT.pgs 3/26/09 12:10 PM Page 269 Answers for SAT Preparation Exercises Chapter 1 1. C 4. E 7. C 10. D 13. B 16. 3 19. 226 Chapter 2 1. C 4. B 7. D 10. C 13. D 16. 15 or 0.2 2. 5. 8. 11. 14. 17. 20. A D A C A 11 28 3. 6. 9. 12. 15. 18. D C B E 17 4 2. 5. 8. 11. 14. 17. C D D E E 7 in. 3. 6. 9. 12. 15. 18. A A B C 28 7 6 21. 196 19. 10 mph 20. 22. 42 boys, 30 girls Chapter 3 1. D 2. 4. D 5. 7. C 8. 10. D 11. 13. C 14. 16. 12 17. 19. 4 20. 22. 40 23. Chapter 4 1. C 2. 4. D 5. 7. B 8. 10. D 11. 13. A 14. 16. 16 17. 19. 24 20. 22. 0.011 23. Chapter 5 1. E 4. E 7. E 10. C 13. A 16. 28 19. 5.5 22. 25 2. 5. 8. 11. 14. 17. 20. 23. Chapter 6 1. D 4. A 7. D 10. B 13. D 16. 9 19. 12 22. 150 Chapter 7 1. A 4. E 7. B 10. A 13. D 16. 36 19. 13 22. 1 9 12 A B C C C 13 62 0 3. 6. 9. 12. 15. 18. 21. 24. D B E A A 5 8 8 D A C A B 8 6 days 2 3 3. 6. 9. 12. 15. 18. 21. 24. B A B B C 6 5 15 A A E C C 1 2.83 100 3. 6. 9. 12. 15. 18. 21. 24. C B E B C 625 8 4 Chapter 8 1. D 4. B 7. C 10. E 13. A 16. 8 19. 0.001 22. 9 Chapter 9 1. B 4. E 7. B 10. D 13. A 16. 100 sq units 19. 26.4° 22. 60.867 Chapter 10 1. E 4. C 7. C 10. C 13. C 16. 9.08 cm 19. 2 22. 55° 269 2. 5. 8. 11. 14. 17. 20. 23. C B D C A 43 70 311 3. 6. 9. 12. 15. 18. 21. 24. B C C E B 63 64 6,860 2. 5. 8. 11. 14. 17. 20. 23. D C E B C 1 0 7 3 3. 6. 9. 12. 15. 18. 21. 24. D D B C D 12 18 8 2. 5. 8. 11. 14. 17. 20. 23. A B A C E 1 5 2 or 2.5 4 3. 6. 9. 12. 15. 18. 21. 24. D B E D D 18 11 1 2. 5. 8. 11. 14. 17. 20. 23. D E C D A 6.6 ft 67° 4 3. 6. 9. 12. 15. 18. 21. 24. D A D A B 0.814 1.333 0.6 2. 5. 8. 11. 14. 17. 20. 23. C A E D D 3. 6. 9. 12. 15. 18. 21. 24. D B A C A 1 10° 5 13 30° 4 16 9 14580AKSAT.pgs 3/26/09 Chapter 11 1. A 4. E 7. C 10. B 13. D 16. 3 19. 0 22. !11 5 Chapter 12 1. D 4. A 7. D 10. E 13. C 16. 1 19. 3 22. 4 5 Chapter 13 1. D 4. C 7. A 10. C 13. D 16. 5p 3 19. 45° 22. 160.5° 2. 5. 8. 11. 14. 17. 20. 12:10 PM D B D C E 30° 1.57 Page 270 3. 6. 9. 12. 15. 18. 21. 23. 2 24. n 2. 5. 8. 11. 14. 17. 3. 6. 9. 12. 15. 18. C E E E E 20 20. 2 !10 5 23. 8 2. 5. 8. 11. 14. 17. 20. 23. E B B D C 213° 45° 36.4° Chapter 14 1. E 4. E 7. C 10. D 13. A 16. 126.9° 19. 79.8 mi 22. 96° Chapter 15 1. A 4. E 7. E 10. B 13. C 16. 3 19. 41 22. 500 Chapter 16 1. A 4. D 7. B 10. E 13. E 16. .3087 19. .8220 22. 56 D C D C C 2 1.41 B A C D A 6 21. 465° 9 24. 64 3. 6. 9. 12. 15. 18. 21. 24. B E E A B 720° 45° 3 270 2. 5. 8. 11. 14. 17. 20. 23. D E B B D 41.4° 157 sq units 9.6 3. 6. 9. 12. 15. 18. 21. 24. B C D A C 49 6.45 25 2. 5. 8. 11. 14. 17. 20. 23. D D C A A 1.19 164 1,727 3. 6. 9. 12. 15. 18. 21. 24. D C C C C 1,900 71 3 10 5 0.3 2. 5. 8. 11. 14. 17. 20. 23. C B E D C 3,360 1,260 0 # n # 100 3. 6. 9. 12. 15. 18. 21. 24. E B D A A .3278 9 7 27 5 0.259 14580AK01.pgs 3/26/09 12:05 PM Page 271 Answers for Textbook Exercises Chapter 1.The Integers 1-1 Whole Numbers, Integers, and the Number Line (page 4) 20. 21. 22. 23. Writing About Mathematics 1. Answers will vary. Example: Have Tina count to three on her fingers, then count to two on her remaining fingers. Show her that if she counts the total number of fingers it equals five. 2. Yes. Both sides of the equation refer to the same distance along the number line. Developing Skills 3. 6 4. 12 5. 5 6. 5 7. 7 8. 7 9. 4 10. 0 11. 0 12. 4 13. 25 14. 2 15. 8 1 (25) 5 3 16. 7 1 (2(22)) 5 9 17. 22 1 (25) 5 27 18. 28 1 (2(25)) 5 23 19. {21, 11} Applying Skills 20. 2$20 21. a. 2$75 b. 2$23 22. 1$100 12n 1 8 5 80, six nights 5 1 3h 5 44, 13 hours 19 1 5d , 49, five plants 5d 1 4 # 14, two hours 1-3 Adding Polynomials (pages 12–13) Writing About Mathematics 1. Yes. If x is negative, 2x 1 1 will always be greater than x. If x is positive (or zero), 2x 1 1 is always greater than x. 2. No. Terms in each polynomial may or may not have like terms in the other polynomial. Furthermore, if like terms have coefficients with equal value but opposite signs, adding them will eliminate a term with that power. Thus, the sum of a trinomial and a binomial may have anywhere from zero to five terms. Developing Skills 3. 5y 2 13 4. 5x2 1 x 1 1 2 5. 7x 2 5x 2 4 6. 3x 7. 2a2b2 1 2 8. 4b2 2 10b 9. 26 2 3b 10. x2 1 7x 2 8 2 11. 4y 2 3y 2 4 12. 2a4 1 a3 2 5a2 1 a 2 1 13. 6 14. 4 15. 25 16. 23 17. x , 12, { . . . , 9, 10, 11} 18. y $ 2, {2, 3, 4, . . . } 19. y # 21, { . . . , 23, 22, 21} 20. c . 4, {5, 6, 7, . . . } 21. 1 22. x # 25, { . . . , 27, 26, 25} Applying Skills 23. $2.00 24. a. 6x 1 10 b. 4 feet wide and 13 feet long 25. 50 cents 1-2 Writing and Solving Number Sentences (pages 8–9) Writing About Mathematics 1. Taking an absolute value always yields a positive number. There is no positive number that can be subtracted from 12 to yield 15. 2. No. Dividing both sides of an inequality by a negative number reverses the direction of the inequality. Developing Skills 3. 7 4. 3 5. 22 6. 22 7. 4 8. 22 9. {9, 213} 10. {25, 11} 11. {21, 7} 12. {10, 213} 13. a . 2, {3, 4, 5, . . . } 14. b $ 4, {4, 5, 6, . . . } 15. 1 , x , 3, {2} 16. 3 , x , 7, {4, 5, 6} 17. 21 $ b $ 2, {21, 0, 1, 2} Applying Skills 18. 156 2 3g # 9, 49 cents 19. 5g 1 3 5 18, three groups 1-4 Solving Absolute Value Equations and Inequalities (pages 16–17) Writing About Mathematics 1. The absolute value of a number is equal to the absolute value of its negative. 271 14580AK01.pgs 3/26/09 12:05 PM Page 272 2. Subtract 7 from both sides. The absolute value is then equal to a negative number, which makes the solution set empty. Developing Skills 3. {27, 17} 4. {22, 214} 5. {21, 6} 6. {23, 7} 7. {1, 7} 8. {3, 24} 9. {5, 9} 10. {3, 23} 11. {2, 210} 12. {23, 8} 13. 14. {23, 17} 15. x , 29 or x . 9, {. . . , 212, 211, 210, 10, 11, 12, . . .} 16. x , 29 or x . 5, {. . . , 212, 211, 210, 6, 7, 8, . . .} 17. 211 # b # 21, {211, 210, 29, . . . , 23, 22, 21} 18. 21 , y , 7, {0, 1, 2, 3, 4, 5, 6} 19. y , 219 or y . 7, {. . . , 222, 221, 220, 8, 9, 10, . . .} 20. b # 21 or b $ 8, {. . . , 23, 22, 21, 8, 9, 10, . . .} 21. 23 , x , 7, {22, 21, 0, 1, 2, 3, 4, 5, 6} 22. The set of integers 23. 0 , b , 10, {1, 2, 3, 4, 5, 6, 7, 8, 9} 24. b , 23 or b . 14, {. . . , 26, 25, 24, 15, 16, 17, . . .} 25. 26. 23 # b # 17, {23, 22, 21, . . . , 15, 16, 17} 27. {253, 254, 255, 256, 257, 258, 259}, 253 # x # 259 28. {150, 151, 152, 153, . . . , 297, 298, 299, 300}, 150 # t # 300 29. c 2 200 # 28, solution 5 172 # c # 228, {172, 173, 174, . . . , 226, 227, 228} 31. a. a 5 c 2 1, b 5 c 2 8 b. a2 1 b2 5 (c 2 1) 2 1 (c 2 8) 2 5 2c2 2 18c 1 65 c. 2c2 2 18c 1 65 1-6 Factoring Polynomials (pages 26–27) Writing About Mathematics 1. Yes. If we multiply these factors back together, we get x2 1 (d 1 e)x 1 de. 2. No. These factors will yield +4 as the last term instead of 24. Developing Skills 3. 4x(2x 1 3) 4. 3a2(2a2 2 a 1 3) 5. 5ab(b 2 3 1 4a) 6. x2y2(xy 2 2x 1 1) 7. 4a(1 2 3b 1 4a) 8. 7(3a2 2 2a 1 1) 9. (y 2 1)(y 1 1) 10. (3b 2 4)(b 2 2) 11. (2x 1 3)(y 1 4) 12. (a2 1 3)(a 2 3) 2 13. (x 2 2)(2x 2 3) 14. (y2 2 5)(y 1 1) 15. (x 1 7)(x 1 1) 16. (x 1 3)(x 1 2) 17. (x 2 3)(x 2 2) 18. (x 1 6)(x 2 1) 19. (x 2 3)(x 1 2) 20. (x 1 4)(x 1 5) 21. (3x 1 4)(x 2 3) 22. (2y 2 1)(y 1 3) 23. (5b 1 1)(b 1 1) 24. (6x 2 1)(x 2 2) 25. (2y 1 1)(2y 1 1) 26. (3x 2 2)(3x 2 2) 27. (a 1 3)(a 1 1)(a 2 1) 28. 5(x 2 1)(x 2 2) 29. b(b 1 2)(b 2 2) 30. 4a(x 1 3)(x 2 2) 31. 3(2c 1 1)(2c 2 1) 32. (x2 1 9)(x 1 3)(x 2 3) 33. (x2 1 4)(x 1 2)(x 2 2) 34. x(2x 1 3)(x 1 5) 35. 2x(2x 2 3)(x 2 1) 36. (z2 2 3)(z 1 3)(z 2 3) 37. (c 1 3)(c 1 1) 38. (y 1 1)(2y 1 3) 5 21(y 2 3)(y 1 1) 39. y(x 1 4)(x 2 4) 40. 3(x 2 3)(x 1 1) 41. 29(x 1 1)(x 1 3) Applying Skills 42. (4x 1 1)(x 2 2) 43. (4x 1 5)(4x 2 5) 44. (3x 2 1)(3x 2 1) 45. (3x 2 1)(x 1 2) 1-5 Multiplying Polynomials (page 21) Writing About Mathematics 1. No. Using FOIL, the answer is (a 1 3)(a 1 3) 5 a2 1 6a 1 9. 2. Six. Each term of the trinomial (3) is multiplied by each term of the binomial (2). Developing Skills 3. 14a8b4 4. 212c3d4 2 4 5. 36x y 6. 9c8 8 7. 29c 8. 15b2 2 12b 2 2 2 3 9. 2x y 2 4x y 10. 2x2 1 5x 2 3 2 11. a 2 a 2 20 12. 3x2 2 5x 2 2 13. a2 2 9 14. 25b2 2 4 15. a2 1 6a 1 9 16. 9b2 2 12b 1 4 3 2 17. y 2 3y 1 3y 2 1 18. 2x3 1 5x2 2 7x 2 15 19. 11a 2 12 20. 4b2 1 5b 2 21. 8y 2 7y 2 10 22. 0 23. z3 2 6z2 1 12z 2 8 24. 25 25. 3 26. 2 27. 1 28. 6 29. 4 Applying Skills 30. 2x2 1 4x 1-7 Quadratic Equations with Integral Roots (pages 29–30) Writing About Mathematics 1. No. If the product of two expressions is zero, at least one of the two expressions must be zero. This is not true for other numbers. 272 14580AK01.pgs 3/26/09 12:05 PM Page 273 2. Yes. If the product of any number of expressions is zero, at least one of the expressions must be zero. Developing Skills 3. {1, 3} 4. 6. {21, 25} 7. 9. {21, 4} 10. 12. {23, 4} 13. 15. {23, 2} 16. 19. {1 ft by 3 ft by 3 ft, 2 ft by 4 ft by 3 ft, 3 ft by 5 ft by 3 ft, 4 ft by 6 ft by 3 ft} Review Exercises (pages 37–38) {2, 5} {2, 212} {3, 210} {1, 7} {1} 5. 8. 11. 14. 17. 1. 3. 5. 7. 9. 11. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35. 36. 37. 38. 39. 40. 41. {21, 6} {21, 10} {22, 3} {3} {5} Applying Skills 18. Francis is 11, Brad is 14. 19. Length: 30 ft, width: 18 ft 20. Width: 8 ft, length: 18 ft 21. 9 cm, 12 cm, 15 cm 22. 3 seconds 1-8 Quadratic Inequalities (page 35) Writing About Mathematics 1. No. If all three factors are negative, the product will be negative. Furthermore, if two factors are negative, the product will be positive. 2. a. Yes. The solution set is 5 , x , 7; thus, any value makes (x 2 7) the negative factor and (x 2 5) the positive factor. b. No. We can tell for binomial factors of the form (x 1 a)(x 1 b) where a and b are given. However, in other products, such as xy, either factor can be the positive factor. Developing Skills 3. 23 , x , 22, [ 4. x , 26 or x . 1, {. . . , 29, 28, 27, 2, 3, 4, . . .} 5. 1 # x # 2, {1, 2} 6. x , 2 or x . 5, {. . . , 21, 0, 1, 6, 7, 8, . . .} 7. 22 , x , 3, {21, 0, 1, 2} 8. x # 22 or x $ 10, {. . . , 24, 23, 22, 10, 11, 12, . . .} 9. 24 , x , 3, {23, 22, 21, 0, 1, 2} 10. x , 1 or x . 5, {. . . , 22, 21, 0, 6, 7, 8, . . .} 11. x # 0 or x $ 2, {. . . , 22, 21, 0, 2, 3, 4, . . .} 12. 22 , x , 3, {21, 0, 1, 2} 13. x , 2 or x . 2, {. . . , 21, 0, 1, 3, 4, 5, . . .} 14. The set of integers 15. 22 , x , 1, {21, 0} 16. 23 # x # 4, {23, 22, 21, 0, 1, 2, 3, 4} 17. x , 23 or x . 4, {. . . , 26, 25, 24, 5, 6, 7, . . .} 42. 43. 44. 45. 22x 2. 2a 1 12 23d 1 7 4. 3b2 2 25b 1 45 2 x 1 7x 2 20 6. 22a2 2 2a 2 2 14d 1 19cd 2 3c 8. x2 2 x 2 1 4 10. 0 2x2 12. y2 2 4y 2(x 1 1)(x 1 3) 3(a 2 5)(a 2 5) or 3(a 2 5) 2 5x(x 1 1)(x 2 4) 10a(b 1 2)(b 2 2) (c2 1 4)(c 1 2)(c 2 2) 3(y2 1 2)(y 2 4) (x 2 1)(x 1 1)(x 1 5) (x 1 1)(x 2 1)(x 1 1)(x 2 1) 2(x 2 3)(x 2 6) x(x 2 2)(x 2 1) 5(a2 1 b2)(a 1 b)(a 2 b) (5x 2 3)(x 1 5) 29 13 x . 4, {5, 6, 7, . . . } 21 # x , 4, {21, 0, 1, 2, 3} {2, 27} {6, 28} y , 21 or y . 2, {. . . , 24, 23, 22, 3, 4, 5, . . .} x , 25 or x . 21, {. . . , 28, 27, 26, 0, 1, 2, . . .} {4, 5} {5, 7} 26 , x , 21, {25, 24, 23, 22} x , 25 or x . 7, {. . . , 28, 27, 26, 8, 9, 10, . . .} 0 # x # 5, {0, 1, 2, 3, 4, 5} x , 23 or x . 0, {. . . , 26, 25, 24, 1, 2, 3, . . .} 1 # x # 3, {1, 2, 3} x # 22 or x $ 1, {. . . , 24, 23, 22, 1, 2, 3, . . .} An absolute value cannot be equal to a negative number. Width: 12 cm, length: 32 cm Width: 8 ft, length: 30 ft 10 in., 24 in., 26 in. a. 96 feet b. 1 second and 4 seconds Exploration (page 38) 1. 6; 28; 496; 8,128 2. All Euclidean perfect numbers have 2k21 as a factor. Since k is always a positive integer greater than 1, Euclidean perfect numbers will be multiples of 2. Applying Skills 18. {1 ft by 2 ft, 2 ft by 3 ft, 3 ft by 4 ft, 4 ft by 5 ft, 5 ft by 6 ft, 6 ft by 7 ft} 273 14580AK01.pgs 3/26/09 12:05 PM Page 274 3. The possible units digits of any power of 2 are {2, 4, 6, 8}. Given the units digit of any 2k21, the units digit of (2k 2 1) 5 2(2k21) 2 1. If 2k21 5 2, then the units digit of (2k 2 1) 5 3. If 2k21 5 4, then the units digit of (2k 2 1) 5 7. If 2k21 5 6, then the units digit of (2k 2 1) 5 1. If 2k21 5 8, then the units digit of (2k 2 1) 5 5. However, integers with units digit 5 (other than 5 itself) are not prime, so (2k 2 1) 5 and 2k21 8. The product of 2 3 3 and of 6 3 1 is 6. The product of 4 3 7 ends in 8. Therefore, a Euclidean perfect number N 5 2k21(2k 2 1) must have a units digit of 6 or 8. Chapter 2.The Rational Numbers 2-1 Rational Numbers (page 43) 7. 9. 11. Writing About Mathematics 1. a. The coin is called a quarter because it is 25 out of 100 cents, one-fourth the value of a dollar. b. A quarter of something is equivalent to onefourth of its total value. Since the total number of minutes in an hour and cents in a dollar differ, one-fourth of those values will also differ. 2. The additive inverse makes the sum of the two numbers equal zero. The multiplicative inverse makes the product of the two numbers equal to one. Developing Skills 8 3 1 8 3. 6. 8. 9. 10. 11. 12. 13. 4. 12 7 7. 1 0.166 . . . 5 0.16 0.222 . . . 5 0.2 0.7142857142 . . . 5 0.714285 0.133 . . . 5 0.13 0.8750 1 14. 23 8 16. 4 11 5 6 7 44 19. 22. 17. 20. 4 37 26 45 13. 15. 17. 19. 21. 22. 24. 26. 5. 27 2 28. 30. a 5 72 c 5 0, 1 3 5 4y x (x 2 0, y 2 0) 3 4c3 (c 2 0, d 2 0) 3y 1 1 2y (y 2 0) 3 2 3 2 4d A d 2 0, 4 B 1 3 1 2xy2 A x 2 0, y 2 2 3 (a 2 25) x 2 4 x 1 5 (x 2 3, 25) 1 3a 1 3 (a 2 21, 1) 1 b 1 2 (b 2 2, 22) 5 2b 1 4 (b 2 24, 4) 8. 10. 12. b 5 2, 23 x 5 21, 0, 6 ab 2 (a 2 0) 14. 2b 3 (b 2 0) 1 4 16. 2a 3a (a 2 0) 2 2b 18. 4a 3a (a 2 0, b 2 0) c 20. c 1 2 (c 2 0, 22) 0, xy2 2 232 B 23. a 2 2 (a 2 22) 25. 5(y 2 2) y 1 2 (y 2 22) 27. a 1 1 (a 2 1) 29. x 22 2 3 (x 2 3) 2-3 Multiplying and Dividing Rational Expressions (pages 52–53) 15. 18. 21. Writing About Mathematics 1. No. Joshua needed to write the reciprocal of the second fraction before attempting to cancel out any common factors. 2. Yes. He divided the terms separately, which is acceptable based upon the commutative property of multiplication. Developing Skills 2 9 47 300 3 22 2-2 Simplifying Rational Expressions (pages 47–48) Writing About Mathematics 1. Abby is wrong. 3x is not a common factor of the numerator and denominator, and cannot be canceled out. 2. No. It is true for all values except where the denominator is zero A a 5 32 B . Developing Skills 3. a 5 0 4. c 5 0 5. a 5 0, b 5 0 6. x 5 25 3. 12 3 4. 28 (a 2 0) 1 5. 10 (x 2 0, y 2 0) 6. 23 (a 2 0) 7. 35 (b 2 21) 2 8. a 16 10a (a 2 0, 10) 9. y2 23 3y (y 2 23, 0, 3) 2(a 2 1) 10. 3(a 1 4) (a 2 24, 22, 4) 11. 1 (a 2 0, 22) 12. 25 2x (x 2 23, 0, 3) 274 13. 53 14. 8 (a 2 0) 15. 4 (b 2 0, c 2 0) 16. 16 (a 2 0) 14580AK01.pgs 3/26/09 12:05 PM Page 275 18. 2y 21 1 A y 2 212, 0, 12 B 1 20. w 1 1 (w 2 21, 0, 1) 3 (x 2 0, 2) 17. 4x 19. 1 (c 2 3) Developing Skills 3. 3 : 2 4. 3 : 2 5. 1 : 6 6. 1 : 5 7. 2 : 3 8. 2 : 3 9. 1 : 3 10. 2 : 7 11. 2 12. 7 13. 7 14. 13 15. 7 16. {23, 2} 17. {0, 5} 18. {21, 4} 19. {210, 2} Applying Skills 20. 16 inches wide by 20 inches long 21. 28 inches long by 12 inches wide 22. 15 games 23. 33 members 24. $75 and $50 25. 214 cups 26. 4 cups of solution and 28 cups of water 1 5 (a 2 0, 23) 22. a 4a 27 2 23. (2x 1 7) (x 2 1) A x 2 1, 2 B 21. b4 (b 2 0, 23) 2 a (a 2 21) 24. a 2 2 26. 9 (x 2 21, 0, 1) 28. (x 2 1)(x 1 4) x 25. 14 27. 12 (a 2 22, 0, 2) (x 2 0, 1) 2 29. 6(b 1 2) (b 2 0) 30. 3x (x 0, 1, 2) 2-4 Adding and Subtracting Rational Expressions (pages 56–57) Writing About Mathematics 1. No. It is also undefined when a 5 1. 2. Yes. He formed a correct LCD and added. Developing Skills 3. x 2 4. 25x5x1 2 (x 2 0) 5. 10x 21 1 9 6. 2a202 9 or 2a 20 7y 1 12 9. x 12x (x 2 0) 12. 2 13. 2a 2a2 3 (a 2 0) 14. 17. 18. 19. 1. {21, 0, 1}. An expression of the form y 1 1 z x 4 z 5 x ? y is undefined when either x, y, or z is zero. 2 2. No. When d2 5 2, the denominator will equal zero, which would make the fraction undefined. Developing Skills 10. 11a6a1 2 (a 2 0) 11. 3x x1 2 (x 2 0) 16. Writing About Mathematics 1 80 8. 3a 40a (a 2 0) 7. 6 15. 2-6 Complex Rational Expressions (pages 63–64) 10y 2 1 (y 2 0) 2y 1 1 x x (x 2 0) x2 1 3x 2 4 x(x 1 2) (x 2 0, 22) 2b 1 1 2(b 2 1) (b 2 1) 1 x 2 2 (x 2 2) 1 (2a 2 1)(a 1 2) A a 2 22, 1 a(a 2 2) (a 2 22, 0, 2) 2 x (x 2 0, 2) 21. a. b. 2 1 6 23. a. 2x x 2 1 3x b. (x 2 1)(x 2 1) 1 4. 10 5. 12 6. 4 (x 2 0) 7. 212, 1 2 9. 22 (d 2 0, 1) 23 B 11. 13. 20. Applying Skills 4x2 1 2 x 3. 4 15. 17. 18x 1 20 3 22. a. b. x 1 1 3 b (b 2 0, 1) 2(2y 1 1) A y 2 0, 12 B y (a 1 7) (a 2 2) (a 2 0, 2, 7) b 1 2 b 2 1 (b 2 21, 0, 1) 19. 21 (a 2 0, 1) 4x2 1 6x 24. a. (x 1 1)(x 1 2) 21. x2 b. (x 1 1)(x 1 1) 23. 1 2x (x 2 0, 21) 3a 1 2 A a 2 0, 53 B 4a 8. a1 (a 2 21, 0) 10. 2(b 1 1) (b 2 0, 1) 12. 12 A y 2 212 B 14. 5x 6 (x 2 0) 3 16. x 2 5 (x 2 0, 3, 5) y 2 2 18. y 1 8 (y 28, 23, 0) 5(a 1 3) a 20. 2 22. 17 4a (a 2 0, 3) (a 2 0) 24. 21 (b 2 22, 0, 2) 2-7 Solving Rational Equations (pages 69–70) 2-5 Ratio and Proportion (pages 60–61) Writing About Mathematics 1. Yes. Interchanging the means or extremes of a proportion maintains the equality of the proportion. 2. Yes. One is added to each side of the equation, which maintains the equality. Writing About Mathematics 1. Yes. Samantha multiplied both sides by the LCD, which is a valid way to solve this equation. 2. Brianna is correct. A rational equation has a variable in one or more denominators. 275 14580AK01.pgs 3/26/09 12:05 PM Page 276 16. 23 Developing Skills 3. 32 4. 8 5. 8 6. 70 7. 12 8. 20 9. 80 10. 20 11. 8 12. 10 13. 12 14. 285 3 1 15. 2 16. U22, 2 V 17. 3 18. {5, 27} 19. 4 20. 73 Applying Skills 21. Week 1: Joseph worked 8 hours and Nicole worked 12. Week 2: Joseph worked 12 hours and Nicole worked 24. 22. 5 mph 23. 40 mph then 50 mph 24. Price: $1.25, 6.6 lb the 1st week, 7.6 lb the 2nd week 17. ba (a 2 0, b 2 0, a 2 2b) 1 18. x 1 6 (x 2 26, 6) 19. 20. a(a 2 5) a 2 4 1 11 (a 2 24, 0, 4) 21. 20 3 22. 6 23. {0, 8} 24. 2 25. U22, 32 V 26. x , 0 or x . 2 9 27. x , 212 or x . 220 28. 29. 30. 31. 27 boys, 30 girls Week 1: 15 cans, week 2: 12 cans, $0.70 per can 60 mph then 45 mph 16 ft by 13 ft and 14 ft by 13 ft Exploration (page 76) 2-8 Solving Rational Inequalities (pages 73–74) 1. 5 Writing About Mathematics 1. The number line must be separated by the solutions to the equation as well as the values at which any of the rational expressions are undefined. 2. 5x:x , 06 . Since the numerator will be positive for any nonzero rational number and undefined at 0, the expression is negative for all x , 0. Developing Skills 3. a , 224 4. y , 8 5. b . 25 6. d , 2 7. a . 153 8. 0 , x , 1 5 9. 0 , y , 4 10. a , 22 or a . 21 11. 35 , x , 4 12. x , 0 or x . 12 13. 27 , x , 25 5 1 1 n 1 1 1 n(n 1 1) n 1 1 n(n 1 1) 1 n 2. 3 4 5 12 1 1 4 or 2 3 5 12 1 1 3. a. 31 1 15 1 c. 21 1 12 14. 25 , a , 21 1 6 2 3 1 5 13 1 14 1 12 b. 12 1 15 d. 12 1 13 1 18 e. 12 1 19 Review Exercises (pages 75–76) 1. 27 2. {21, 0, 1} Cumulative Review (pages 77–78) 3. 0.416 2a 4. 3b (a 2 0, b 2 0) Part I 1. 4 2. 1 4. 2 5. 1 7. 2 8. 2 10. 1 Part II 11. Answer: 2 # x # 5 7 2 2x # 3 7 2 2x # 3 and 7 2 2x $ 23 x$2 x#5 5. 13 20a (a 2 0) 6. 2x(x32 4) (x 2 24, 0, 4) 2 7. 5a a1 25 (a 2 25, 22, 0) 8. 2a2 1 1 (a 2 21, 1) 10. 12. 14. a 2 1 d 2 3 d (d 2 26, 0) 2a 1 2 a2 1 2a (a 2 22, 21, 0) x 2 1 x 2 2 (x 2 22, 21, 2) y 1 6 9. y 1 3 (y 2 23, 3) 11. 12 (b 2 0) 13. a 1 4 (a 2 4) 1 15. 2x 1 x (x 2 0, 1) 276 3. 3 6. 1 9. 3 14580AK01.pgs 12. 3/26/09 12:05 PM Page 277 2 2 9 a 2 3 3 4 a 15 a 1 5 2 5 3 a 2 3 15 5a 1 ? (a 2 3)(a 5 2 5 1 3) 3 3 5a 1 2 a 1 3 5 3(a 1 3) 2 3(a 1 5) 5 (a 1 5)(a 1 3) 26 5 (a 1 5)(a 1 3) (a 2 25, 23, 3) Part IV 15. Answer: x , 21 or x . 72 2x2 2 5x . 7 2 2x 2 5x 2 7 . 0 (2x 2 7)(x 1 1) . 0 The solutions for the corresponding equations are 27 and 21. When x , 21, the inequality is true. When 21 , x , 72 , the inequality is false. Part III 13. 5x 1 5 x2 2 x 15 x2 2 1 ? 5(x 1 1) x(x 2 1) 5 (x 2 1)(x 1 1) ? 15 5 x3 (x 2 21, 1) When x . 72 , the inequality is true. 16. Diego traveled at 60 mph and then at 20 mph. 14. Width: 6 m, length: 15 m l 5 w2 1 12 w2 2 30 x 1 10 x 51 3 w A w2 1 12 B 5 90 30 x 1 12w 2 90 5 0 2 w 1 24w 2 180 5 0 (w 1 30)(w 2 6) 5 0 w 5 230, w 5 6. Reject negative value. 1 30 x 5 1 60 x 5 1 x 5 60 Chapter 3. Real Numbers and Radicals 3-1 The Real Numbers and Absolute Value (page 83) 3-2 Roots and Radicals (pages 87–88) Writing About Mathematics 1. a. Yes. Since the product of an even number of either positive or negative factors is positive, the radical will have both a positive and a negative root. b. Yes. The product of an odd number of positive factors is positive, and the product of an odd number of negative factors is negative. Thus a radical with a real, nonzero radicand and an odd index will have only one real root. 2. a. Yes. This is true only when a $ 0. The other equal factor is 2!a. b. The statement is true for a $ 0. When a , 0, a has no square roots in the set of real numbers. Developing Skills 3. rational 4. irrational 5. neither 6. rational 7. rational 8. rational 9. rational 10. rational 11. 4 12. 64 13. 24 14. 25 15. 13 16. 20.2 17. 60.8 18. 1.2 19. 3 20. 2 21. 25 22. 25 23. 5 24. 21 25. 25 7 2 26. 26 27. 3 28. 212 29. 0.1 30. 0.4 31. x3 Writing About Mathematics 1. No. The expression can be written as the ratio A 154 B of two integers, so it is rational. 2. No. Maria’s inequality is a false statement. If she applied the rule “x . k, then x . k or x , 2k,” she would get 22x 1 5 . 3 or 2x 2 5 , 23. Developing Skills 3. rational 5. irrational 7. rational 9. irrational 11. rational 13. rational 4. 6. 8. 10. 12. 14. irrational irrational irrational rational irrational irrational In 15–26, answers will be graphs of number lines. 15. 27 , x , 7 16. a $ 8 or a # 2 17. y . 2 or y , 27 18. 21 # b # 2 19. a , 29 or a . 21 20. 24 , x , 2 21. x . 25 or x , 215 22. { } or 23. all real numbers 24. x 5 24 5 25. all real numbers 26. all real numbers Applying Skills 27. 70° # t # 220° 28. 282 ft # h # 20,320 ft 277 14580AK01.pgs 3/26/09 32. 10c2 2 35. 2b6 38. x2 41. x $ 23 44. a 5 614 Applying Skills 47. 48. 49. 50. 51. 33. 36. 39. 42. 45. 12:05 PM 0.5x 20.1y x$2 x $ 25 b 5 610 Page 278 34. 37. 40. 43. 46. 10 a Applying Skills 39. 4!13 cm 40. 6!2 in. 41. 12!3 m 42. 5!13 ft 43. 5!6 ft 44. xy2 !5 m 2 45. 6y units 46. The longest diagonal of the trunk is !1,604 or approximately 40.05 inches. Thus, everything but the walking stick will fit. 1 x#3 x 5 69 y 5 613 !14 cm 3.74 cm !544 ft 23.32 ft 15 in. 6 in. 13 ft 3-4 Adding and Subtracting Radicals (pages 97–98) Writing about Mathematics 1. Yes, for x . 0. (3x)2 5 9x2, and "9x2 5 3x. Her 3-3 Simplifying Radicals (pages 93–94) Writing About Mathematics substitution is correct. 2. No. We do not add radicands. !16 1 !48 5 4 1 4!3, which is not equal to !64 5 8. Developing Skills 1. 2!36 is the negative of the square root of 36, which is a real number, simplifying to 26. !236 is the square root of a negative number and is not real. 2. Negative. If a is negative, 28a3 will be positive and its cube root will be also positive. The negative sign in front makes the whole expression negative. Developing Skills 3. 2 !3 4. 5 !2 7. 7c2 !2 8. 6y2 !5y 5. 4 !2 9. 50y !2x 11. 3b2 !2ab 3 13. 2 ! 3 3 15. 5xy2 " 3x2 2 17. 2a5 6 19. " 9y3 a 21. 5b !5ab 23. 25. !3xy 2y !10a 6 2a 27. 9b 2 !3ab 29. 5y2 2 !10xy 31. !2 2 x 33. 100 !10x 35. 10 !3c a2 "2b2c bc 4 37. 12. 5a3 !2a 14. 12x!2x 15. 11b !6b 16. 22x3 !5 17. 6 !5 5 19. 4!7 18. 3!6 20. !2x x 21. 5a!5 2 5!2a 23. 6 !3y 1 y 18. b7 !3b 2 2 20. a2 !2a 25. 9!3 2 2!6 27. 12 !6 !6xy 22. 6xy 3 29. 7! 2 31. 8!x !15b 5b3 !30b 4b2 3 10xy4 !2xy !2 5 11b2 !2ab 10 33. !2a 22. 22x!6 24. 6a2b !2b 1 2 26. 3!5 2 !10 10 3 28. 3! 2 4 30. ! 3 32. 5!y 34. 23b !2 35. 3a!7 2 3a!5 or 3aA !7 2 !5B 36. b !a 37. 15x!2x 38. 3x!x 40. 34. 8x !2 36. 10. 5b2 !11 2 4 16. 2a2 " 3ab3 32. 8. !5y 13. 4c2 !2c 3 14. 2a ! 5a 30. 6. 4!7 11. 5y !6x 3 12. 2 ! 2 28. 5. 9!3 9. 6a!10 10. 44x2y3 !3xy 26. 4. 2!5 7. 6!2 6. 2b !2b 24. 3. 6!2 42. a 3 3 !9 !2 2 !5 3 Applying Skills 4 38. 2xy ! 2x 43. 14 !3 in. 278 39. !3 41. !6 14580AK01.pgs 3/26/09 12:05 PM Page 279 44. 25 !3 ft 3-6 Dividing Radicals (pages 103–104) 45. a. 5 !2 cm Writing About Mathematics b. 12 !2 cm 46. a. b. 47. a. b. 1. No. Jonathan’s error was treating the !10 denominator of !10 2 as !2. 2 does not simply 14 in. 14!2 1 14 in. or 14A1 1 !2B in. 34 !10 m 13 !10 m further. 5 !9 5 3, 2. Answers may vary. Example: !27 !3 5 !2, which is irrational. which is rational. !6 !3 3-5 Multiplying Radicals (pages 100–101) Developing Skills Writing About Mathematics 1. Yes. !2 is a positive real number. 2. Yes. We can simplify by dividing the exponent of the radicand by the index. Developing Skills 3. 4 7. 26 !5 11. 4 !10 13. 27 19. 21. 12. 12 19. !2 2 3!10 24. 3 27. 12 !2 1 4 28. 5a 2 3 !5a 23. 1 1 35. 236 2 !6 2 37. a 2 b 39. 9 1 6b !5ab 1 5ab 3 18. 2 1 4 !6 20. 2 1 !3 22. 5 1 2 !6 3 2 4 29. 8!x 1 ! 40 24. !5 1 6 3 4 3 26. a1" a or "aa 3 w 28. 2 ! w 30. 22 1 2!5 30. 52 !7 ft 31. a. 2!6 cm b. 6 cm 4 34. 2x2 2 3x ! 3y 1 !3y 3-7 Rationalizing a Denominator (pages 107–108) 36. 6 2 36c2 Writing About Mathematics 38. 4 2 2 !3 1. a. If Juan writes 7 as !49, the fraction becomes !49 . This 2 !7 40. 26 2 6 !7 is equivalent to 12#49 7 , which simplifies to 12 !7. 7 b. No. Juan’s procedure cannot be applied to 2 !5 because 5 is not a factor of 49. 2. Brittany took the fraction at face value and multiplied by the conjugate of the denominator. Justin saw that the denominator factored to 2A1 1 !2B . 2 is a factor of the numerator, so the fraction is equivalent to 1 12 !2 . 43. 120 ft2 b. 6 1 2!6 ft 4 Applying Skills Applying Skills 42. 4,608 m2 44. !5 in.3 45. a. 2 !6 ft 2 !6xy 3y !3x x 16. !3a 27. !c 32. 21 2 4 !5y 2 5y 33. 49 2 5b 14. 25. !9 1 "12x 3 26. !5 2 5 !2 31. 9 1 10!2b 1 2b 41. 22 2 !5 21. 1 x 2 !x 3 23. 3a2 ! 5 12. 17. c !2 22. 2 29. 6xy2 1 6y!3xy 10. 2!3 a !14 2 3 !2b 2b !7y y 15. 20. 25. 2 !2 1 2 8. 5!2 10. 2 !7 18. x3y2 !3 35a 3 a 6 !3a 7. 13. 16. 4ab !b 17. 2y2 !5 6. a!10 2 !2x x 6. 4 !6 14. 20 15. 2x2 5. 3 11. 8. 6 !5 9. 2 !2 4. 5 9. 3!3 4. 15 5. 9 3. 2 c. 3 ft2 46. p(4 1 xy5 1 4y2 !xy) m2 279 14580AK01.pgs 3/26/09 12:05 PM Developing Skills Page 280 3. !3 3 4. 12 !10 7. !3 !6 12 5. 2 !2 8. 19. 14. 1 3 2 !5 4 !3 2 1 2 3 !5 2 3 4 16. 5 123!2 18. 6 1 32 !3 20. 16 29 4 !7 21. 3!5 1 6 22. 3!7 2 6 23. !2 1 1 24. 3 2 !3 1 2 !5x 25. 5x5x 2 4 27. 8 1 75 !2 26. 28. 29. 11 2 92 !10 30. 31. 32. 33. 35. 36. 37. 38. 39. Writing About Mathematics 1. There is no real number for which the square root is negative. If x $ 23 the radicand will not be negative, so there will be a solution in the set of real numbers. 2. No. Once we square the equation it has two real roots. One is the root of the given equation and the other is the root of !15 2 7x 5 1 2 x. Developing Skills 3. 25 4. 49 5. 9 6. 36 7. 16 8. 32 9. 4 10. 8 11. 4 12. 44 13. 4 14. 2 15. 2 16. 25 17. 4 18. 4 19. 5 20. 2 21. 21 22. 5 23. 5, 4 24. { } (no solution) 25. 5 26. 3 27. 1 12. 16 !6 13. 61 !10 (a 1 2) Ab 1 !2B 2 b 2 2 4 !z 2 32 z 2 64 2y !x 1 2x !y xy 34. 3 !x 1 (36 2 x) !6 2 18 3x 2 108 2a !ab 1 2b !ab or 2 !ab ab ab (a 1 3x2 2 3x !2 1 (5x2 2 10) !x x(x2 2 2) 2 a. 3 !6 10y !y 2 2 !5y 5y2 2 1 17 1 2 !7 9 27 1 7 !5 22 2x 1 5y 2 7 !xy x 2 y a 1 2 !a a 2 4 30. 8 31. 210 32. 22 33. 234 34. 18 35. 4 36. {0, 8} 37. 5 38. 5 Applying Skills 39. 8 units each 40. Width 5 2, length 5 1 41. a. AB 5 BC 5 2!10, AC 5 !10 b. 5!10 b) Review Exercises (pages 114–116) In 1–8, answers will be graphs of number lines. 40. a. 2 !33 1 3 or 23 !3 1 1 b. 2.154700538 c. 2.154700538 41. a. 2 !3 1 2 b. 5.464101615 c. 5.464101615 1. 214 , x , 34 2. 26.5 # x # 2.5 3. 210.5 # x # 11.5 4. x , 22 or x . 4.5 1 5. x # 23 5 or x $ 25 6. x . 1 or x , 213 7. 21 # x # 215 9. 8!2 42. a. 8 1 3 !7 b. 15.93725393 c. 15.93725393 45. !3 2 1 29. U1, 23 V 28. 3 b. 1.632993162 c. 1.632993162 43. !2 48. 32 !2 ft, 32 !2 ft, 2!2 ft 3-8 Solving Radical Equations (pages 112–113) 10. !3 11. 23 !6 17. 47. (6!5 2 6) in. 6. 23 !3 9. 43 !3 15. Applying Skills 44. !3 280 10. !3 2 11. 15!3 12. !5 13. 21!3 14. 4!2 15. 5!6 1 2!5 16. 12 17. 75 46. !8 2 !5 8. 0 , x , 2 18. 4 1 3 !2 14580AK01.pgs 3/26/09 12:05 PM Page 281 19. 20 2 20 !2 20. 6!2 1 60 21. 21 22. 22 23. 5 1 3 !3 2x2 2 9x 2 5 5 0 24. !2 (2x 1 1)(x 2 5) 5 0 26. 4 !2 2 2 25. 2!5 1 1 27. 4 !3131 3 29. 7b !2b x 5 212 28. 8 !a Check: x 5 2 32. 12x2 !x ? 5 2 A 14 B 1 92 5 34. x y !y 4 2 2 33. 4x 2 x!36 35. 3a ! 2 3 3 38. 3912!x x 40. a 39. 16 116a 22 a8 !a 41. x2 42. 13 43. 5 44. 18 45. 14 46. {3, 4} 47. 5 48. 25 49. {22, 3} 50. 12 1 7!3 ft 2x 1 4 x 51. a. 4 m b. 4 1 4 !2 m x2 1 A !2x 1 1B 2 5 (x 1 1) 2 x2 1 2x 1 1 5 x2 1 2x 1 1 True for all values of x . 0 because of the radical. 53. a. 24 2 12 !3 ft b. 52 2 22 !3 ft Cumulative Review (pages 116–118) 5 3. 4 6. 3 9. 4 w(4w 1 2) 5 30 a2 5 5 16 60 5 16 60 16x 5 480 x 5 30 Rachel travels 30 mph on local streets and 60 mph on the highway. 16. let w 5 the width in yards. The length is therefore l 5 4w 1 2. Use A 5 lw. Answers will vary. 5 5 83 2 1 6 x 8 x Exploration (page 116) 2. 3 5. 1 8. 4 Check: x 5 5 ? 5 2(5) 2 2 9(5) 5 ? 5 2(25) 2 45 5 5 5 5✔ 8x 5 6x 1 12 2x 5 12 x56 Plugging back in, Tyler answered 16 questions correctly. Part IV 15. Use D 5 RT, where D is distance, R is speed, and T is time. Then T 5 D R. Let x 5 Rachel’s speed on local streets, and 2x be her highway speed, both in mph. 16 2 12 x 1 2x 5 60 52. Show that these values satisfy the Pythagorean Theorem: Part I 1. 4 4. 3 7. 4 10. 1 Part II x55 5 5 5✔ Part III 13. 3b 1 6 , 7 27 , 3b 1 6 , 7 213 , 3b ,1 213 1 , b , 3 3 1 1 243 , b ,3 14. No. correct 5 2(no. incorrect) + 4 Let no. incorrect 5 x No. correct 5 2x + 4 36. 4a2 3 37. b2 ! 9 212 ? 5 2 A 212 B 2 9 A 212 B 5 4 30. x ! 12x 31. !ab b 11. 2x2 2 9x 5 5 12. 2 16 a2 2 a 2 12 a2 2 16 a2 2 a 2 12 4 a2 4w2 1 2w 2 30 5 0 1 4a 2a 2w2 1 w 2 15 5 0 2a ? a2 1 4a (a 1 4)(a 2 4) (a 1 3)(a 2 4) 2 a 1 3 ? (2w 2 5)(w 1 3) 5 0 2a a(a 1 4) w 5 212 w 5 23 Reject negative value. 1 The garden is 22 yards wide and 12 yards long. 281 14580AK02.pgs 3/26/09 12:05 PM Page 282 Chapter 4. Relations and Functions 4-1 Relations and Functions (pages 126–127) 14. a. {all real numbers} b. The function is not onto since the range is {y : y $ 0}. 15. a. {all real numbers} b. The function is not onto since the range is Uy : y # 1 4 V. Writing About Mathematics 1. {(x, y) : x 5 y2} is not a function because for all values of x . 0 there are two distinct y-values, whereas 5(x, y: !x 5 y6 is a function because for every value of x $ 0 there is exactly one real number that is the square root. No two pairs have the same first element. 2. No, because not every positive integer has an integral square root. The range contains noninteger values. Developing Skills 3. a. function; no two pairs have the same first element b. {1, 2, 3, 4} c. {1, 4, 9, 16} 4. a. not a function; points (1, 21) and (1, 1) have the same first element b. {0, 1} c. {21, 0, 1} 5. a. function; no two pairs have the same first element b. {22, 21, 0, 1, 2} c. {5} 6. a. function b. {all real numbers} c. {y : y $ 21} 7. a. not a function b. {all real numbers} c. {y : y # 21 or y $ 1} 8. a. not a function b. {x : x $ 22} c. {all real numbers} 9. a. function b. {all real numbers} c. range: y 5 2 10. a. function b. {x : 23 # x # 3} c. {y : 0 # y # 4} 11. a. function b. {x : 1 # x # 6} c. {y : 0 # y # 2.5} 12. a. {all real numbers} b. The function is not onto since the range is {2183}. 13. a. {all real numbers} b. The function is onto since the range is equal to the domain. 16. a. {x : x $ 0} b. The function is onto since the range is equal to the domain. 17. a. {all real numbers} b. The function is not onto since the range is {y : y $ 0}. 18. a. {x : x 0} b. The function is onto since the range is equal to the domain. 19. a. {x : x # 3} b. The function is onto since the range is equal to the domain. 20. a. {x : x . 21} b. The function is not onto since the range is {y : y . 0}. 21. a. {all real numbers} b. The function is not onto since the range is {y : 0 , y # 1}. 22. a. {x : x 1} b. The function is onto since the range is equal to the domain. 23. a. {x : x 3} b. The function is not onto since the range is {y : y , 1}. Applying Skills 24. a. {(x, y) : y 5 x(6 2 x)} b. y 5 x 25 O 1 2 3 4 5 6 7 8 9 10 210 215 220 225 230 235 240 x 0 1 2 3 4 5 6 7 8 9 10 y 0 5 8 9 8 5 0 27 216 227 240 c. {x : 0 , x , 6} 25. a. {(x, y) : y 5 10x} b. (0, 0), (1, 10), (2, 20), (3, 30), (4, 40), (5, 50), (6, 60), (7, 70), (8, 80) c. {0, 1, 2, 3, 4, 5, 6, 7, 8} d. {0, 10, 20, 30, 40, 50, 60, 70, 80} 282 14580AK02.pgs 3/26/09 12:05 PM Page 283 4-2 Function Notation (pages 128–129) Developing Skills 3. a. {1, 2, 3, 4} b. {4, 7, 10, 13} c. yes 5. a. {2, 3, 4, 5, 6} b. {7} c. no 7. no 8. no 10. yes 11. no 13. a. graph b. yes c. yes 15. a. graph b. yes c. yes 17. a. graph b. yes c. yes Writing About Mathematics 1. f and g are the same function, since evaluating g at x yields the same value as f. 2. f and g are not the same function. For example, note that f(3) 5 9, while g(3) 5 g(1 1 2) 5 3. Developing Skills 3. a. f(x) 5 x 2 2 4. a. f(x) 5 x2 b. 3 b. 25 5. a. f(x) 5 3x 2 7 6. a. f(x) 5 5x b. 8 b. 25 7. a. f(x) 5 !x 2 1 8. a. f(x) 5 x2 b. 25 b. 2 9. 10 10. 10 12. 4 13. 2 15. a. 2 b. 23 Applying Skills 16. a. t(a) 5 0.08a b. {a : a $ 0} c. $0.40 d. $1.32 17. a. y 900 800 700 600 500 400 300 200 100 2100 2200 2300 2400 2500 2600 2700 2800 2900 c. 22 11. 12 14. 1 d. 2 19. a. 4. a. b. c. 6. a. b. c. 14. a. b. c. 16. a. b. c. 18. a. b. c. {0, 2, 4, 6} {8, 6, 4, 2} yes {0, 21, 22, 23, 24} {3, 5, 7, 9, 11} yes 9. no 12. no graph no yes graph no yes graph no yes y 1 x O1 b. yes c. yes x O 20. a. y 1,000 2,000 3,000 1 x O1 b. $300 c. 3,000 muffins 4-3 Linear Functions and Direct Variation (pages 133–135) Writing About Mathematics 1. Yes. g(x) 5 a1f(x) , when a . 1, is equivalent to f(x) 5 ag(x), and ag(x) is the graph of g(x) stretched vertically by a factor of a. 2. Yes. Directly proportional means the ratio of r : s is constant. Every direct variation of two variables is a linear function that is one-to-one. b. yes c. yes 21. no 22. yes Applying Skills 23. nc 5 6 24. dt 5 35 26. 27. g k 28. a. b. c. d. e. 283 5 1,000 g m 25. fi 5 12 5 25 g(t) 5 80 1 25t {t : 0 # t # 420} {g(t) : 80 # g(t) # 10,580} yes No. The ratio g(t) : t is not constant. 14580AK02.pgs 3/26/09 12:05 PM Page 284 Hands-On Activity 1 1–2. y h(x) Hands-On Activity 2 1–2. y g(x) 1 g(x) 1O 1 x 21 O x h(x) 3. The graph of p(x) is the graph of f(x) shifted a units to the left. 4. The graph of p(x) is the graph of f(x) shifted a units to the right. 5–6. y f(x) 1 2 3. The graph of g(x) is the graph of f(x) reflected in the y-axis. 4. The graph of h(x) is the graph of f(x) reflected in the x-axis. 5. y x O 21 1 1 O 1 x f(x) 2 4 7. The graph of f(x) 1 a is the graph of f(x) shifted a units up. 8. The graph of f(x) 2 a is the graph of f(x) shifted a units down. 9. y 6. The graph of p(x) is the graph of f(x) stretched vertically by a factor of 3. 7. The graph of 2p(x) is the graph of f(x) reflected in the x-axis and stretched vertically by a factor of 3. 8. f(x) 5 x 1 1 (1) Graph of g(x) 5 f(2x) 5 2x 1 1 (2) Graph of h(x) 5 2f(x) 5 2x 2 1 (3) The graph of g(x) is the graph of f(x) reflected in the y-axis. (4) The graph of h(x) is the graph of f(x) reflected in the x-axis. (5) Graph of 3f(x) 5 3x 1 3 (6) The graph of p(x) is the graph of f(x) stretched vertically by a factor of 3. (7) The graph of 2p(x) is the graph of f(x) reflected in the x-axis and stretched vertically by a factor of 3. 1 O x 1 10. The graph of af(x) is the graph of f(x) stretched vertically by a factor of a. 11. y O 21 1 284 x 14580AK02.pgs 3/26/09 12:05 PM Page 285 17. a. 12. The graph of 2f(x) is the graph of f(x) reflected in the x-axis or y-axis. 13. y y 1 21 O O 21 1 x x b. {28, 2} c. x , 28 or x . 2 18. a. y O 21 1 x 14. The graph of g(x) is the graph of f(x) reflected in the x-axis or y-axis. 4-4 Absolute Value Functions (pages 138–139) d. b. c. d. Writing About Mathematics 1. Yes. Each y then corresponds to exactly one x. 2. Yes. By definition, f(x) 5 2 2 x when x # 2 and f(x) 5 x 2 2 when x . 2. Developing Skills 3. (0, 0) 4. (24, 0) 5. (14, 0) 6. (5, 0) 4-5 Polynomial Functions (pages 147–149) Writing About Mathematics 1. Yes. Tiffany is correct. The graph never crosses or touches the x-axis. 2. This function has two roots; x 5 2 is a double root. Developing Skills 3. a. no real roots 4. a. 21 b. {3} b. {all real numbers} c. no c. yes d. no d. yes 5. a. {0, 2} 6. a. {21, 3} b. {y : y $ 21} b. {y : y # 4} c. no c. no d. no d. no 7. a. 1 8. a. 0 b. {y : y # 0} b. {all real numbers} c. no c. yes d. no d. yes 9. a. {22, 0, 2} 10. a. {23, 21, 1, 3} b. {all real numbers} b. {y : y $ 21} c. no c. no d. no d. no 11. a. {22, 0, 3} b. {all real numbers} c. yes d. no 12. 23 , x , 1 13. 23 # x # 21 14. x , 21 or x . 2 15. { } or 16. x , 1 or x . 5 17. 24 # x # 1 In 7–10, the answer to part a is a graph. 7. b. {y : y $ 0} 9. b. {y : y $ 1} Applying Skills 11. h(x) 5 2 2 x 12. a. m(x) 5 x 2 150 8. b. {y : y $ 0} 10. b. {y : y $ 23} x 2 150 13. 14. 15. 16. 28 , x , 2 {2, 6} 2,x,6 x , 2 or x . 6 b. h(x) 5 65 a–c. Graphs d. The graph of y 5 x 1 a is the graph of y 5 x shifted vertically by the amount a. When a . 0, the shift is upward. When a , 0, the shift is downward. a–c. Graphs d. The graph of y 5 x 1 a is the graph of y 5 x shifted a horizontally. When a . 0, the shift is to the left. When a , 0, the shift is to the right. a–b. Graphs c. The graph of y 5 2x is the graph of y 5 x reflected in the x-axis. a–c. Graphs d. The graph of y 5 ax is the graph of y 5 x stretched or compressed vertically. When a . 0, the graph is stretched vertically. When a , 0, the graph is compressed vertically. 285 14580AK02.pgs 3/26/09 12:05 PM Page 286 Applying Skills 18. a. 12 2 x b. y 5 x(12 2 x) or y 5 12x 2 x2 c. y 36 33 30 27 24 21 18 15 12 9 6 3 25. Writing About Mathematics 1. No. 2 2 x and x 2 2 are both always $ 0 so their sum is always $ 0. Their sum is equal to 0 only when x 5 2. 2. If g(x) 5 x 1 1, 2g(x) 5 2x 1 2 g(2x) 5 2x 1 1. If f(x) 5 x, 2 f(x) 5 2x 5 f(2x) 5 2x. Developing Skills 3. a. {0, 1, 2, 3, 4, 5} b. {1, 2, 3, 4, 5, 6} c. {1, 2, 3, 4, 5} d. {(1, 23), (2, 1), (3, 7), (4, 15), (5, 25)} e. {1, 2, 3, 4} f. U A 1, 14 B , A 2, 43 B , A 3, 92 B , (4, 16) V 2 4 6 8 10 d. length 5 width 5 6 19. a. 20 2 x 2 b. y 5 12x(20 2 x) or y 5 10x 2 x2 c. y 55 50 45 40 35 30 25 20 15 10 5 25 2 4-6 The Algebra of Functions (pages 153–155) x O A 22ab , 4ac4a2 b B 4. a. f(0) 5 1 b. x 5 1 c–e. y 2f(x) O O 1 x 1 f(x) 1 2 x 2f(x) 5 10 15 20 25 d. Each leg measures 10 feet. 20. p(x) 5 (x 1 4)(x 1 2)(x 2 3) or p(x) 5 x3 1 3x2 2 10x 2 24 21. a–c. Graphs d. y 5 x2 1 a is the graph of y 5 x2 shifted vertically by the amount a. When a . 0, the graph of y 5 x2 is shifted upward. When a , 0, the graph of y 5 x2 is shifted downward. e. T0,a 22. a–c. Graph d. y 5 (x 1 a)2 is the graph of y 5 x2 shifted horizontally by the amount a. When a . 0, the shift is to the left. When a , 0, the shift is to the right. e. Ta,0 23. a–b. Graphs c. The graph of y 5 2x2 is the graph of y 5 x2 reflected in the x-axis. d. Ry=0 24. a–c. Graphs d. y 5 ax2 is the graph of y 5 x2 stretched vertically. e. y 5 ax2 is the graph of y 5 x2 compressed vertically. 5. a. f(0) 5 22 b. x 5 62 c–e. y f(x) 1 2 x 21 O 21 2f(x) 2f(x) 6. a. f(0) 5 1 b. { } or c–e. y f(x) 1 2 1 21 O 2f(x) 286 x 2f(x) 14580AK02.pgs 3/26/09 12:05 PM 7. a. f(0) 5 21 b. x 5 61 c–e. Page 287 12. a. h(x) 5 3x 2 1 b. {all real numbers} c. {all real numbers} 13. a. h(x) 5 4 1 x2 b. {all real numbers} c. {y : y $ 4} 14. a. h(x) 5 x2 1 8x 1 16 b. {all real numbers} c. {y : y $ 0} 15. a. h(x) 5 x b. {x : x $ 0} c. {y : y $ 0} 16. a. h(x) 5 22 1 x b. {all real numbers} c. {y : y # 0} 17. a. h(x) 5 5 2 x b. {all real numbers} c. {y : y $ 0} 18. a. h(x) 5 x b. {all real numbers} c. {all real numbers} 19. f(g(x)) 5 x 1 3 g(f(x)) 5 x 1 3 20. f(g(x)) 5 2x g(f(x)) 5 2x 21. f(g(x)) 5 2x 1 3 g(f(x)) 5 2x 1 3 22. f(g(x)) 5 5 2 x g(f(x)) 5 5 2 x 23. Exercise 20: g(x) 5 2x 24. p(q(5)) 5 2 q(p(5)) 5 4 25. Answers will vary. For example, f(x) 5 2x and g(x) 5 x 1 1. Applying Skills 26. a. c(x) 5 1.08x b. d(x) 5 x 2 10 c. c + d(x) 5 1.08x 2 10.80 d + c(x) 5 1.08x 2 10 No. c + d(x) applies sales tax to the discounted price, while d + c(x) discounts the price after sales tax has been applied. It makes sense to use c + d(x) on in-store discounts. It makes sense to use d + c(x) on discounts applied after purchase, for example, a $10 mail-in rebate. d. The function to use depends on whether the tax is applied to the full price or to the discounted price. 27. a. n + f 5 4(0.55x2 1 1.66 x 1 50) 2 160 5 2.2x2 1 6.64x 1 40 b. 128.2 chirps per minute y f(x) 1 2 2f(x) O x 2f(x) 8. a. x2 2 2x 1 4 b. {all real numbers} 10. a. x 9. a. x2 2 x1 b. {x : x 0} 2 11. a. 4 2x 2x b. {x : x 2} 13. a. 2x3 2 4x2 b. {all real numbers} b. {x : x 0} 12. a. x2 2 6x 1 12 b. {all real numbers} Applying Skills 14. a. c(x) 5 10x 1 2 b. t(x) 5 0.15(10x) 5 1.5x c. e(x) 5 10x 1 2 1 0.15(10x) 5 11.5x 1 2 d. e(3) 5 $36.50 15. a. c(x) 5 8.50x b. s(x) 5 0.50x 1 2 c. t(x) 5 8.50x 1 0.50x 1 2 5 9x 1 2 d. t(5) 5 $47.00 16. Answers will vary. Example: Let f(x) 5 x 1 1; then show f(x) 1 f(x) 5 2f(x). f(x) 1 f(x) 5 x 1 1 1 x 1 1 5 2x 1 2 2f(x) 5 2(x 1 1) 5 2x 1 2 This result is true in general since doubling a function is the same as adding it to itself. 4-7 Composition of Functions (pages 159–160) Writing About Mathematics 1. Yes. f(x) 5 x2 evaluates to (a 1 1)2 at x 5 a 1 1 by definition. 2. fg(x) is the product of the functions f(x) and g(x). f(g(x)) is function composition. Developing Skills 3. 6 4. 10 5. 212 6. 28 7. 45 8. 1 9. 24 10. 0 In 11–18, the answer to part d is a graph. 11. a. h(x) 5 8x 1 4 b. {all real numbers} c. {all real numbers} 287 14580AK02.pgs 3/26/09 12:05 PM Page 288 4-8 Inverse Functions (pages 166–167) 25. Writing About Mathematics 1. Yes. The graph of the inverse is the reflection of the graph of f over the line y 5 x. 2. No. When the domain of an absolute value function is the set of real numbers, the function is not one-to-one and has no inverse function. Developing Skills 3. 3 4. 5 5. 22 6. 8 7. 12 8. 26 9. 26 10. !2 11. Yes; f21 5 {(8, 0), (7, 1), (6, 2), (5, 3), (4, 4)} 12. Yes; f21 5 {(4, 1), (7, 2), (10, 1), (13, 4)} 13. Yes; f21 5 {(8, 0), (6, 2), (4, 4), (2, 6)} 14. No. The function is not one-to-one, so it has no inverse function. 15. No. This relation is not a function. 16. Yes; f21 5 {(x, y): y 5 !x 2 2 for 2 # x # 27} 3 17. a. f21 (x) 5 x 1 4 b. domain of f 5 domain of f21 5 {all real numbers} range of f 5 range of f21 5 {all real numbers} 18. a. g21(x) 5 x 1 5 b. domain of g 5 domain of g21 5 {all real numbers} range of g 5 range of g21 5 {all real numbers} 19. a. f21(x) 5 3x 2 5 b. domain of f 5 domain of f21 5 {all real numbers} range of f 5 range of f21 5 {all real numbers} 20. a. f21 5 x2 b. domain of f 5 domain of f21 5 {x : x $ 0} range of f 5 range of f21 5 {y : y $ 0} 21 21. f 5 U (x, y) : y 5 x5 V 1 26. O 1 x O 21 1 x y Applying Skills x 27. f21 5 0.2532 x f(f21 (x)) 5 0.2532 A 0.2532 B 5x 0.2532x 21 f (f(x)) 5 0.2532 5 x Since f(f21(x)) 5 f21(f(x)) 5 x, the functions are inverses. 28. a. Graph b. domain 5 {x : x $ 24} range 5 {y : y $ 2} c. domain 5 {x : x $ 2} range 5 {y : y $ 24} d. The domain of the function is the range of the inverse and the range of the function is the domain of the inverse. 22. g21(x) 5 7 2 x. Yes, a function can be its own inverse. 23. No. y 5 x2 is not one-to-one if the domain is the set of real numbers. 24. y 1O 21 y 4-9 Circles (pages 172–173) Writing About Mathematics 1. No. A circle does not pass the vertical line test. 2. In center-radius form, the constant term is the square of the radius, and this cannot be negative. Developing Skills 3. a. x2 1 y2 5 4 b. x2 1 y2 2 4 5 0 4. a. x2 1 y2 5 9 b. x2 1 y2 2 9 5 0 5. a. x2 1 y2 5 16 b. x2 1 y2 2 16 5 0 6. a. (x 2 4)2 1 (y 2 2)2 5 1 b. x2 1 y2 2 8x 2 4y 1 19 5 0 7. a. (x 1 1)2 1 (y 2 1)2 5 16 b. x2 1 y2 1 2x 2 2y 2 14 5 0 x 288 14580AK02.pgs 3/26/09 12:05 PM Page 289 8. a. (x 2 6)2 1 (y 2 5)2 5 100 b. x2 1 y2 2 12x 2 10y 2 39 5 0 9. a. (x 2 6)2 1 (y 2 13)2 5 169 b. x2 1 y2 2 12x 2 26y 1 36 5 0 10. a. x2 1 (y 2 1)2 5 17 b. x2 1 y2 2 2y 2 16 5 0 11. x2 1 y2 5 16 12. (x 2 2)2 1 (y 2 3)2 5 1 13. (x 2 1)2 1 (y 1 1)2 5 9 14. (x 1 2)2 1 (y 2 3)2 5 4 15. (x 2 1)2 1 (y 1 1)2 5 25 16. x2 1 (y 1 1)2 5 4 17. (x 1 1)2 1 (y 2 3)2 5 9 18. (x 2 1)2 1 (y 2 1)2 5 13 19. (x 1 1)2 1 (y 1 1)2 5 13 20. a. x2 1 y2 5 25 b. (0, 0) c. 5 21. a. (x 2 1)2 1 (y 2 1)2 5 9 b. (1, 1) c. 3 22. a. (x 1 1)2 1 (y 2 2)2 5 4 b. (21, 2) c. 2 23. a. (x 2 3)2 1 (y 1 1)2 5 16 b. (3, 21) c. 4 24. a. (x 1 3)2 1 (y 2 3)2 5 12 b. (23, 3) c. 2 !3 25. a. x2 1 (y 2 4)2 5 16 b. (0, 4) c. 4 26. a. (x 1 5)2 1 (y 2 2.5)2 5 63.25 b. (25, 2.5) c. !63.25 5 !253 2 27. a. b. A x 1 12 B A 212, 32 B 2 31. Answers will vary: any three equations of the form (x 2 2)2 1 (y 2 2)2 5 r2 with three different positive values for r. Hands-On Activity 1. (1, 3) 2. Slope of PQ 5 0; the slope of the line perpendicular to PQ is undefined. 3. x 5 1 4. (1) (0, 0) (2) Slope of QR 5 21; slope of the line perpendicular to QR 5 1 (3) y 5 x 5. C(1, 1) 6. CP 5 CQ 5 CR 5 2!5 7. Equation: (x 2 1)2 1 (y 2 1)2 5 20 ? P is on the circle: (5 2 1)2 1 (3 2 1)2 5 20 16 1 4 5 20 ✔ ? Q is on the circle: (23 2 1)2 1 (3 2 1)2 5 20 16 1 4 5 20 ✔ ? R is on the circle: (3 2 1)2 1 (23 2 1)2 5 20 4 1 16 5 20 ✔ 4-10 Inverse Variation (pages 177–178) Writing About Mathematics 1. No. The function f cannot be represented in the form f(x) 5 y 5 anxn 1 an21xn21 1 c 1 a0. 2. In direct variation, both quantities increase or decrease by the same factor. In inverse variation, as one quantity increases by a factor a, the other quantity decreases by the factor a1 . Developing Skills 3. xy 5 2 or y 5 x2 4. xy 5 6 or y 5 x6 1 A y 2 32 B 5 18 4 5. xy 5 28 or y 5 28 x 6. inversely 7. directly 8. directly 9. directly 10. inversely 11. neither directly nor indirectly 12. directly Applying Skills 13. The width of rectangle ABCD is equal to half the width of rectangle EFGH. 14. He can ride to school in one-third the time it takes him to walk. 15. a. Yes. D 5 RT. As rate increases, time traveled decreases, when distance is constant. b. 45 mph 2 c. 32 !2 Applying Skills 28. a. Graph b. Yes. The cube can easily pass through the arch because its sides are shorter than the radius of the arch. c. Yes. If the prism is placed in the center of the arch so that its base is 8 feet, it will have just under 7 feet of clearance to pass under the arch. 29. 10 !2 30. Width 5 4 !5, length 5 8 !5 289 14580AK02.pgs 3/26/09 12:05 PM Page 290 16. Initial trip: 55 mph for 3 hrs Return trip: 33 mph for 5 hrs 17. 1st time: 16 cans at $1.50 per can 2nd time: 15 cans at $1.60 per can 33. 34. 35. 36. 37. Review Exercises (pages 180–183) 1. Not a function since (1, 1) and (1, 21) have the same first element. 2. Function. Each x of the domain has only one y-value. 3. Not a function since for most values of x in the domain there are two distinct y-values in the range. 4. Function. Each x value of the domain has only one y-value in the range. 5. Function. Each x value of the domain has only one y-value in the range. 6. a. yes 7. a. no b. yes b. no 3 c. yes; y 5 x 2 4 8. a. no b. no c. no 10. a. no b. no c. no 12. 14. 16. 18. 19. 20. 21. 38. 40. 42. c. no 9. a. yes b. yes 3 c. yes; y 5 ! x 43. 11. a. yes b. yes c. yes; y 5 x2 1 4, x $ 0 44. 1 13. {21, 3} {22, 4} 15. {22, 21, 1} 1 17. {23, 5} x 5 27 and x 5 1 {(0, 7), (1, 8), (2, 9), (3, 10), (4, 11)} {(0, 5), (1, 2), (2, 21), (3, 24), (4, 27)} {(0, 6), (1, 15), (2, 20), (3, 21), (4, 18)} 45. Exploration (page 183) 1. 2. 3. 4. 5. 22. U (0, 6), A 1, 53 B , A 2, 45 B , A 3, 37 B , (4, 29 B V 23. f(g(x)) 5 2x 1 6 g(f(x)) 5 2x 1 3 25. f(g(x)) 5 (x 1 2)2 g(f(x)) 5 x2 1 2 27. f(g(x)) 5 2 12 3x g(f(x)) 5 2 1 32x 29. f(g(x)) 5 x g(f(x)) 5 x (x 2 3)2 1 y2 5 25 x2 1 (y 1 1)2 5 16 (x 1 1)2 1 (y 1 1)2 5 5 (x 1 2)2 1 (y 2 2)2 5 8 (21, 22) and (3, 2) Check (21, 22): (x 2 2) 2 1 (y 1 1) 2 5 10 ? 10 (23) 2 1 (21) 2 5 10 5 10 ✔ y5x21 ? 21 2 1 22 5 22 5 22 ✔ Check (3, 2): (x 2 2) 2 1 (y 1 1) 2 5 10 ? 10 (1) 2 1 (3) 2 5 10 5 10 ✔ y5x21 ? 321 25 2 5 2✔ y 5 (x 2 4)2 2 2 39. y 5 3x y 5 22x 2 3 41. y 5 2x 1 1 1 3 a. f21(x) 5 21 (x 2 8) b. Yes, since f is one-to-one. a. f21 (x) 5 x3 b. Yes, since f is one-to-one. a. f21 (x) 5 !x b. No, f is not one-to-one. c. {x : x $ 0} a. f21(x) 5 x2 b. Yes, since f is one-to-one. 24. f(g(x)) 5 4x 2 1 g(f(x)) 5 4x 2 1 The base is a circle. The two cut edges are circles. The cut surfaces are ellipses. The curved portion of the edges is a parabola. The shape is a hyperbola. The shape is a pair of intersecting lines. Cumulative Review (pages 184–185) 26. f(g(x)) 5 !5x 2 3 g(f(x)) 5 5 !x 2 3 Part I 1. 3 2. 2 4. 2 5. 3 7. 4 8. 2 10. 4 Part II 11. 2(x 1 1)2 (x 2 1) 1 28. f(g(x)) 5 2x g(f(x)) 5 50 x 30. f(g(x)) 5 x g(f(x)) 5 x 3. 2 6. 4 9. 4 A3 1 !5B 2 !5 3 1 !5 12. 33 1 ? 5 9 2 5 5 7 1 23 !5 2 !5 3 1 !5 31. x2 1 y2 5 9 32. (x 2 3)2 1 (y 2 3)2 5 9 290 14580AK02.pgs 3/26/09 12:05 PM Page 291 Part III 13. Answer: 23 , x , 5 x2 2 2x 2 15 , 0 (x 1 3)(x 2 5) , 0 Let (x 1 3) , 0: x13,0 x25.0 x , 23 x.5 Solution: { } Let (x 1 3) . 0: x13.0 x25,0 x . 23 x,5 Solution: 23 , x , 5 Combine the two solutions. Part IV 15. a. y 1O 1 x b. x 5 1 c. (1, 29) d. x 5 20.5 and x 5 2.5 14. D 5 22h 5 24, E 5 22k 5 6 F 5 h2 1 k2 2 r2 5 4 1 9 2 16 5 23 x2 1 y2 1 Dx 1 Ey 1 F 5 0 x2 1 y2 2 4x 1 6y 2 3 5 0 16. 212 a 1 2 12 a 2 2 2a(a 1 1) 2a 2a ? aa2 5 2aa2 1 2 1 5 (a 1 1)(a 2 1) 5 a 2 1 ; undefined for a 5 0, a 5 1. Chapter 5. Quadratic Functions and Complex Numbers 5-1 Real Roots of a Quadratic Equation (pages 192–193) 10. a. Writing About Mathematics 1. 0 5 2x2 2 x 2 1 0 5 16x2 2 8x 2 8 0 5 8(x 2 1)(2x 1 1) x 5 1, 212 1 2. Yes. The resulting equation is equivalent to the original. The new equation can be solved by completing the square. Developing Skills 3. 19 5 (x 1 3)2 4. 116 5 (x 2 4)2 5. 11 5 (x 2 1)2 6. 136 5 (x 2 6)2 7. 12 5 2(x 2 1)2 8. 194 5 Ax 2 b. 20.7, 2.7 y c. 1 6 !3 O 1 11. a. x y b. 20.6, 23.4 c. 22 6 !2 1 3 2 2B O 1 x In 9–14, part b, answers will vary. 9. a. y b. 0.8, 5.2 c. 3 6 !5 1 21O 12. a. b. 1.3, 4.7 c. 3 6 !3 y x 1 O 1 291 x 14580AK02.pgs 3/26/09 13. a. Page 292 1 Writing About Mathematics 1. No. The denominator applies to all the terms in the numerator. 2. Yes. When b2 , 4ac, the roots involve the square root of a negative number, which is not real. Developing Skills 3. 21, 24 4. 27, 1 5. 3 62 !5 x x 7. 25 62 !33 10. 21 6 !5 6. 1 6 2!17 9. 0, 3 b. 2.4, 7.7 c. 5 6 !7 y O 1 5-2 The Quadratic Formula (pages 195–197) b. 20.4, 2.4 c. 1 6 !2 y 1O 1 14. a. 12:05 PM 12. 1 6 8!17 15. 3 6 !6 13. 5 6 4!33 16. 12 6 !3 18. a. y 8. 62!2 11. 32, 1 14. 1 6 4!33 17. 2 6 3!10 1 O1 x 15. 1 6 !3 16. 23 6 !5 19. 3 6 !7 20. 4 6 2!3 17. 2 6 !3 21. 23 6 #15 2 23. 3 62 !3 25. 21 6 !7 18. 21 6 !6 22. 1 6 2 !3 3 24. U12, 92 V 26. 23 62 !33 b. Answers will vary: 20.4, 25.6 c. 23 6 !7 d. 20.4, 25.6 27. a. 5 6 2!21 b. 0.2, 4.8 28. translated 6 right, 31 down 29. translated 1 left, 3 down 30. translated 3 right, 16 down 31. translated 21 left, 2 up 32. reflected about x-axis, translated 21 right, 241 or 9 4 up 33. stretched vertically by a factor of 3, translated 1 unit left 34. Vertex: (24, 211), axis of symmetry: x 5 24 Complete the square to get f(x) 5 (x 1 4)2 2 11. Applying Skills 19. 20. 21. 22. 23. 24. 1 1 !6, 7 1 2 !6 or 1 2 !6, 7 2 2 !6 Width 5 21 1 !3 ft, length 5 1 1 !3 ft Width 5 22 1 !46 cm, length 5 2 1 !46 cm Altitude 5 23 1 3!5 ft, base 5 3 1 3!5 ft Bases 5 8, 12; height 5 4 DB 5 22 1 2!37, AD 5 2 1 2!37, AB 5 4!37 25. a. b. A 2b 2 !b2 2 4ac , 2a A 2b 2a , 0 B 2 2 4ac 0 B and A 2b 1 !b , 0B 2a c. x 5 2b 2a Applying Skills d. 2b 2a 35. a. Width 5 21 1 2 !5 ft, length 5 2 1 4 !5 ft b. A21 1 2 !5B A2 1 4!5B 5 2A2!5 1 1B A2!5 2 1B 5 38 ft2 c. Width 5 3.5 ft, length 5 10.9 ft 36. a. Base 1 5 2 !6 2 2 ft, base 2 5 2 !6 1 6 ft, height 5 2 !6 2 2 ft 26. 11.9 seconds 27. a. 1.2 b. 2.3 c. 16.7 2 2 b b c 28. x2 1 bax 1 4a 2 5 4a2 2 a b. Q 2 !62 2 2 R A A2 !6 2 2B 1 A2 !6B B A x 1 2ab B 5 A !6 2 1B A4!6 1 4B 5 20 ft2 c. Base 1 5 height 5 2.9 ft, base 2 5 10.9 ft 37. Steve is 13, Alice is 15. Use x(x 1 2) 5 195. 2 2 4ac 5b 2 4a2 2 2 4ac x 5 2b 6 !b 2a The roots are the same. 292 14580AK02.pgs 3/26/09 12:05 PM Page 293 Hands-On Activity: Alternate Derivation of the Quadratic Formula Yes. 25. a. c 5 1 b. any c , 1 such that 4 2 4c a perfect square c. any c , 1 such that 4 2 4c is not a perfect square d. c . 1 26. a. 64 b. any b , 24 or b . 4 such that b2 2 16 is a perfect square c. any b , 24 or b . 4 such that b2 2 16 is not a perfect square d. 24 , b , 4 Applying Skills 27. The fence cannot be constructed. Use x2 1 (15 2 x) 2 5 82. The discriminant is 2188, so the equation has no real roots. 28. Yes. Use 4x(5 2 x) 5 25. The discriminant is 0. 29. Yes. Use 216x2 1 48x 5 32. The determinant is 256. 30. No. That value for the profit yields a negative determinant. 2ax 1 b 5 6"b2 2 4ac 2ax 5 2b 6 "b2 2 4ac 2 x 5 2b 6 "2ab 2 4ac 5-3 The Discriminant (pages 201–203) Writing About Mathematics 1. a. 9 b. 2!52 6 3 c. No. The rules apply only when a, b, and c are rational numbers. 2. Yes. Since b2 is always positive, when 24ac is positive, b2 2 4ac . 0. Developing Skills 3. , 0 4. . 0 5. , 0 6. 5 0 7. . 0 8. 5 0 9. a. rational and unequal b. 2 10. a. irrational and unequal b. 2 11. a. rational and equal b. 1 12. a. irrational and unequal b. 2 13. a. not real numbers b. 0 14. a. rational and unequal b. 2 15. a. 0, rational and equal b. 6 16. a. 49, rational and unequal b. 0, 272 17. a. 5, irrational and unequal b. 18. a. b. 19. a. 5-4 The Complex Numbers (pages 208–209) Hands-On Activity For the parallelogram with vertices 4 1 2i, 2 2 5i, and 0, the fourth vertex is 6 2 3i, which is the sum of the two given complex numbers. In 1–9, the resulting complex number is always the sum of the two complex numbers. Student answers should include graphs of parallelograms on the complex plane. 1. 5 1 5i 2. 27 1 7i 3. 6 2 4i 4. 21 2 7i 5. 2 6. 210 7. 23 2 4i 8. 3i 9. 4 1 2i Writing About Mathematics 1. No. Factoring out i from each term and then multiplying yields the product 24. 2. Yes. i ? i 5 i2 5 21 and any real coefficient, when squared, is positive. Developing Skills 3. 2i 4. 9i 5. 3i 6. 26i 7. 211i 8. 2i !2 9. 2i !3 10. 26i !2 11. 15i !3 12. 22i !5 13. 2i !51 14. 10i !5 15. 5 1 i !5 16. 1 1 i !3 17. 24 2 2i !6 18. 23 1 6i 19. 19i 20. 3i 21. 13i 22. 3i 23. 0 24. 7i !5 23 6 !5 2 64, rational and unequal 62 17, irrational and unequal b. 1 6 8!17 20. a. 211, not real numbers b. no real roots 21. a. 0, rational and equal b. 12 22. a. 49, rational and unequal b. 21, 52 23. a. 211, irrational and unequal b. no real roots 24. Yes. A perfect square trinomial is the only way to yield equal rational roots. 293 14580AK02.pgs 25. 27. 29. 31. 33. 35. 37. 39. 41. 43. 45. 3/26/09 24 1 8i 2 1 6i !2 21 2 4i!10 23 1 7i 21 1 i!6 i 6i i 1 i (22, 5) 12:05 PM 26. 28. 30. 32. 34. 36. 38. 40. 42. 44. 46. Page 294 15 !2 1 6i !2 24 1 i !7 5 2 3i i !2 3 7 2 4 0.45 1 0.3i 23 25i 7 2 2i 0 (2, 4) (4, 22) 27. 29. 31. 33. 35. 48. 49. (0, 3) 51. (0, 0) Applying Skills 50. (23, 0) 52. 13i ohms 53. 212i ohms 1 3. 23 2 1 2i 5. 210i 4. 4i 6. 6 2 9i Writing About Mathematics 1. Yes, i 2 5 21. 2. Yes, (a 1 bi)(a 2 bi) 5 a2 2 b2i 2 5 a2 1 b2. Developing Skills 13. 83 2 76 i 28 15. 215 4 1 5 i 16. 29 4 1 20. 28 2 i 25. p 2 2i 7 4i 53. 45 1 35 i 55. 1 2 4i 20 54. 21 29 2 29 i 56. 3 2 4i 7 1 1 125 i 57. 125 6 59. 4p2 1 1 1 4p12p 2 1 1i 58. 2i 1 i 60. 215 1 35 62. 21 5 i 3. 2 6 2i 4. 23 6 i 5. 2 6 3i 1 6. 21 2 6 2i 8. 24 6 i 9. 1 6 3i 10. 12 6 i 212 12. 2 6 i 6 2i 13. 2 6 i !3 14. 1 6 i !2 5-7 Sum and Product of the Roots of a Quadratic Equation (pages 223–224) 121 10 i 19. 2 1 5i 23. 24 2 50. 15 1 35 i 52. 75 2 11 5 i 11. 12. 34 2 14i 14. 45 1 25 i 18. 3 2 4i 1 3i 49. 3 2 4i 7. 25 6 2i 5 2 4i 24 2 2i 10i 22 2 19i 13 17. 25 42 2 24 i 21. 26 1 9i 48. 4 2 3i Writing About Mathematics 1. Yes. b2 2 4ac will be negative, and since b 5 0, there will be no real component. 2. Yes. The two roots are made by adding and subtracting the imaginary component from the same real component. Developing Skills 5-5 Operations With Complex Numbers (pages 215–216) 4. 6. 8. 10. 46. 6 2 2i 5-6 Complex Roots of a Quadratic Equation (page 219) 5 2i 2. 2 42. 85 1 45 i 30 44. 349 2 108 349 i 47. 21 1 3i 61. 43 1 6i 6. 2 2 3i 3. 3 1 8i 5. 22i 11. 0 1 1 6 1 6i 8 2 17 1 17 i 9 p 81 1 p2 2 81 1 p2 i Applying Skills 4. 5 1. 8 1 16i 7 1 9i 9 2 4i 216i 7i 40. 215 2 25 i 45. Multiplication by a real number 3. 5. 7. 9. 1 39. 10 2 15 i 51. 214 2 25 2 25 2 12i 11 1 23i 17 34i 1 38. 12 2 12 i 43. Hands-On Activity: Multiplying Complex Numbers Multiplication by i 1. 23 1 2i 2. 212 28. 30. 32. 34. 36. 37. 21 41. A 12, 4 B 47. (24, 22) 3 1 15i 2 2 23i 25i 2148 1 Writing About Mathematics 1. x2 2 2px 1 p2 2 q 5 0 2. Both. Olivia’s equation is Adrien’s multiplied by 2. Developing Skills 3. Sum 5 21 4. Sum 5 24 Product 5 1 Product 5 5 22. 12 1 3i 24. 53 1 23i 26. 29 1 7i 294 14580AK02.pgs 3/26/09 12:05 PM 5. Sum 5 32 Page 295 6. Sum 5 225 Product 5 21 Product 5 22 7. Sum 5 2 Product 5 43 8. Sum 5 3 Product 5 21 5-8 Solving Higher Degree Polynomial Equations (pages 227–228) 10. Sum 5 21 9. Sum 5 8 Product 5 212 Product 11. Sum 5 25 5 294 Writing About Mathematics 1. Yes. This follows from the definition of a root. 2. Yes. f(x 2 a) is a translation of f a units to the right. Thus, each root is increased by a. Developing Skills 3. 0, 22, 25 4. 0, 1, 22 5. 23, 6 2i 6. 1, 6i!3 12. Sum 5 0 Product 5 214 Product 5 24 13. Sum 5 0 Product 5 1 14. Sum 5 22 Product 5 0 15. Sum 5 234 16. Sum 5 22 Product 48. c is the product of the roots. Since 2b is an integer equal to the sum of the roots and one root is an integer, both roots are integers. Therefore, both roots are factors of c. 5 298 Product 5 23 7. 9. 11. 13. 17. Sum 5 21 Product 5 35 18. 210 19. 11 20. 252 21. 213 4 24. 1 22. 216 23. 65 25. 211 4 26. 3 61 61, 62 63, 63i 61, 63 15. 0, 63!2 21 6 i !2 3 8. 10. 12. 14. 28. a. 3 2 !2 b. 6 2 9 !2 c. The coefficients of the equation are not rational numbers. 29. a. 211 2 !3 b. 3 1 11 !3 c. The coefficients of the equation are not rational numbers. 31. x2 2 11x 1 28 5 0 30. x2 2 7x 1 10 5 0 2 32. x 2 x 2 12 5 0 33. x2 1 3x 1 2 5 0 2 34. x 2 9 5 0 35. 4x2 2 16x 1 7 5 0 36. 32x2 2 12x 2 9 5 0 37. x2 2 x 5 0 38. x2 2 4x 1 1 5 0 39. x2 2 x 2 1 5 0 2 40. 9x 1 6x 2 2 5 0 41. x2 2 6x 1 10 5 0 2 42. 4x 2 12x 1 13 5 0 43. 4x2 1 9 5 0 Applying Skills 44. x2 2 15x 1 54 5 0 45. x2 2 12x 1 40 5 0 46. Answers will vary. Correct as long as 2b 5 c. Example: x2 2 4x 1 4 5 0 47. Sum: 3, 6 !2 2 i 6i, 62i 612, 612 i 0, 21, 2 16. 21, 12, 3 62i !7 17. 61, 18. 19. a. 0 20. b. Yes 21. a. 2 22. b. No 23. a. 1 24. b. No 25. a. 0 26. b. Yes 27. a. 1 1 9i 28. b. No Applying Skills 29. a. Multiply out to check. 27. 257 2b 1 "b2 2 4ac 2a 3 2, 22, 61 a. 216 b. No a. 0 b. Yes a. 0 b. Yes a. 3 1 !3 b. No 159 a. 233 4 1 8 i b. No b. 1, 212 6 !3 2 i c. Same as part b. The two equations are equal, so they have equal roots. d. Prove by multiplying. 30. a. Multiply out to check. b. 21, 12 6 !3 2 i c. Same as part b. The two equations are equal, so they have equal roots. d. Prove by multiplying. 31. a. The graph of g(x) is that of f(x) stretched vertically by a factor of 2. b. They are the same. c. The graph of p(x) will be that of q(x) stretched vertically by a factor of a. d. They are the same. Hands-On Activity a. 21, 1, 2 b. 22, 2, 3 c. 22, 21, 3 d. 22, 21, 1, 2 2 2b 1 2b 2 "2ab 2 4ac 5 22b 2a 5 a Product: 2b 1 "b2 2 4ac 2b 2 "b2 2 4ac ? 2a 2a b2 2 (b2 2 4ac) c 5 5 a 4a2 295 14580AK02.pgs 3/26/09 12:05 PM Page 296 5-9 Solutions of Systems of Equations and Inequalities (pages 236–239) 45. a. Writing About Mathematics 1. The solutions of 0 . ax2 1 bx 1 c are the x-coordinates of the solutions of y . ax2 1 bx 1 c that are also on the x-axis. 2. The minimum value of x2 1 2 is (0, 2). The range is y $ 2 and never intersects y 5 22. Developing Skills 3. (21, 3) and (3, 3) 4. (24, 4) and (2, 22) 5. (1, 5) and (3, 1) 6. (1, 4) and (4, 1) 7. (2, 2) and (4, 4) 8. (1, 2) and (22, 21) 9. (21, 2) and (4, 7) 10. (21, 21) and (1, 3) 11. (2, 2) and (3, 1) 12. (1, 4) and (4, 7) 13. (20.5, 2.5) and (0, 3) 14. (0.5, 2.5) and (3, 5) 15. (0.5, 2.25) and (4.5, 6.25) 16. (28, 26) and (3, 5) 17. (3, 6) 18. (0, 0) and (5, 15) 19. (21, 23) 20. (21, 2) and (22, 3) 21. (2, 26) and (8, 6) 22. (0.5, 2) and (2, 5) 23. (0.5, 3.25) and (4, 5) 24. (0.5, 4.5) and (3, 7) 25. (22.5, 23.25) and (2, 21) 26. (22.5, 30) and (3.5, 18) 27. no real common solutions b. yes y 1 1 O 46. a. x O 1 2 3 4 5 6 7 8 9x 235 236 237 238 239 240 241 242 243 y 47. a. 232221 O 1 2 3 4 x b. no b. yes 213 214 215 216 217 218 219 28. A2!2, 6 2 4 !2B and A !2, 6 1 4 !2B 29. A2!3, 7 2 !3B and A !3, 7 1 !3B 30. A1 2 !2, 5 2 2 !2B and A1 1 !2, 5 1 2 !2B y 31. A2 2 !5, 28 2 11 !5B and A2 1 !5, 28 1 11 !5B 48. a. y b. no 32. (1.4, 15) and (0, 1) 33. (4 1 2i, 4 2 2i) and (4 2 2i, 4 1 2i) 34. (4, 2) and (2, 24) 8 !5 4 !5 8 !5 35. Q24 !5 5 , 2 5 R and Q 5 , 5 R 36. y 5 22x, y 5 x2 2 2 37. y 5 2x 1 5, y 5 2(x 2 2)2 1 5 1 O 38. (1) 39. (4) 40. (2) 41. (3) 42. (6) 43. (5) 44. a. 21 1 O 49. a. 24232221O 1 2 3 4 x 25 26 27 28 29 210 211 212 b. no y x 1 x y 296 b. yes 14580AK02.pgs 3/26/09 50. a. 12:05 PM Page 297 y b. yes Review Exercises (pages 241–243) b. no 4. 7. 10. 13. 16. 19. 22. 1. i 1 O 21 51. a. x y 2 O 2 x 2i !3 24i 0 1 3i 24 2 2i 0 1 6i 80 2 18i 1 1 0i 3. 3i 7i 4 !6 3 4 1 0i 10 2 4i 17 1 11i 6 1 12i 12i 6. 9. 12. 15. 18. 21. 24. 7i !2 72i 0 1 0i 0 2 10i 12 1 16i 2 2 32i 2 1 0i 26. 52 2 15 i 27. 215 1 8i 28. 5 2 12i 29. 30. 4, 23 37. 1 62 !5 40. 32 6 i 1,400,000 1,300,000 1,200,000 1,100,000 1,000,000 900,000 800,000 700,000 600,000 500,000 400,000 300,000 200,000 100,000 5. 8. 11. 14. 17. 20. 23. 12 25. 25 13 1 13 i 31. 22 6 i 34. 27.5, 4 Applying Skills 52. Width 5 6 ft, length 5 8 ft 53. 6 m, 7 m 54. 4 ft by 8 ft 55. a. (x 2 4)2 1 (y 2 2)2 5 20 b. Graph c. (0, 4) and (6, 22) 56. a. Graph b. Yes. The graphs intersect. c. A3 2 !6, 8 2 2!6B and A3 1 !6, 8 1 2 !6B 57. a. Graph b. No. The graphs do not intersect. c. x 5 1 6 2i, y 5 2 6 4i 58. (22, 23), (0, 1), (2, 5) 59. 0.4 , t , 2.8 60. a. x 2 4 b. V 5 2(x 2 4)2 c. x . 12 61. a. y 2. 4i 43. 210 6 20i 46. 61, 63 49. 50. 51. 52. 53. 54. 55. 57. 59. 61. 63. 7 6 3 !5 2 32. 3 6 !19 35. 1 6 i 38. 1 62 !3 41. 2, 61 44. 223, 56 47. 24, 6 !3 3 33. 3 6 i 36. 0.5, 22 39. 1 65 !6 42. 61, 62 45. 0, 3 6 !5 48. 26, 253 , 1 translated 1 left, 1 up translated 23 left, 734 up scaled by 4, translated 34 right, 43 up reflected x-axis, scaled by 2, translated 54 right, 678 down a. real, rational, unequal b. real, irrational, unequal c. No. The parabola crosses the x-axis in two distinct real points. Yes, the discriminant is positive. (2, 0) and (23, 5) 56. (1, 0) and (4, 3) (0, 0) and (5, 10) 58. (3, 4) and (4, 3) (6, 0) and (3, 3) 60. (21, 4) and (1, 2) (22, 6) and (3, 1) 62. (22, 24) and (4, 8) (21, 24) and (3, 0) 64. (3, 24) and (6, 12) 65. Q 7 2 2!29, 2 2 !29R and Q 7 1 2!29, 2 1 !29R 66. Q 21 22 !5, 1 22 !5 R and Q 21 12 !5, 1 12 !5 R 67. (20.5, 1.75) and A 53, 5 B 68. 69. 70. 71. 72. (1 2 i, i) and (1 1 i, 2i) (22 2 i, 25 2 2i) and (22 1 i, 25 1 2i) (2, 1) and (21, 22) x2 2 2x 2 15 5 0 x2 1 72x 2 2 5 0 or 2x2 1 7x 2 4 5 0 73. x2 2 5 75. x2 2 12x 1 40 5 0 77. 24 , b , 4 10 0 20 0 30 0 40 0 50 0 60 0 70 0 80 0 90 1,0 0 00 x b. more than 20 and less than 1,060 297 74. x2 2 10x 1 7 5 0 76. 9 78. c # 94 14580AK02.pgs 3/26/09 79. a. 12:05 PM Page 298 b. No y 1 O 1 80. a. x b. Yes y 1 21 O 2. c. Yes d. Roots 5 23, 21, 2; factors 5 (x 1 3), (x 1 1), (x 2 2) 3. c. Yes d. Roots 5 1, 3; factors 5 (x 2 1), (x 2 1), (x 2 3) 4. c. Yes d. Roots 5 1, 2; factors 5 (x 2 1), (x 2 1), (x 2 2) Cumulative Review (pages 244–246) x Part I 1. 3 4. 1 7. 3 10. 1 2. 2 5. 2 8. 2 81. 8.3 m by 11.7 m Part II 82. a. i 3 1 i 7 1 11. 23 2 2 i ? 3 1 i 5 10 2 10 i 3 12. 6 ? x 2 5 2x 31 1 ? 6 2 100 90 80 70 60 50 40 30 20 10 y 3. 2 6. 3 9. 1 3x 2 9 5 4x 1 2 x 5 211 Part III !5 3 1 !5 13. 33 1 ? 5 14 14 6 !5 5 7 1 23 !5 2 !5 3 1 !5 14. 3 2 2x 5 0 2x 5 3 2x 5 23 2x 5 3 x 5 232 x 5 32 x O 2 4 6 8 10 12 14 1618 20 22 b. $2 or $22 c. The maximum profit is $100 when the price is $12. 83. The shorter side must be longer than 10 inches. Part IV 15. a–b. y Exploration (pages 243–244) 1. (x 2 1)(x 2 2)(x 1 2) 2. (x2 1 x 2 2)(x 2 2) 3. 1 21 24 4 — 2 2 2 24 1 1 22 0 4. They are the same. 1 O 21 In (1)–(4), parts a and b, answers will vary depending on the choice of root used. 1. c. Yes d. Roots 5 1, 2, 3; factors 5 (x 2 1), (x 2 2), (x 2 3) x 16. a. f + g(23) 5 2((23) 2) 1 4 5 22 b. h(x) 5 2x2 1 4 298 14580AK02.pgs 3/26/09 12:05 PM Page 299 Chapter 6. Sequences and Series 6-1 Sequences (pages 250–252) 45. 1, 1, 2, 3, 5, 8, 13, 21, 34, 55 Hands-On Activity 1. a. 3 b. 7 c. 15 2. an+1 5 2an 1 1, a1 5 3 Writing About Mathematics 1. Randi. Unless an upper limit is defined, the sequence is infinite. 2. a. Yes. an11 5 3(n 1 1) 2 1 5 3n 1 2 5 (3n 2 1) 1 3 5 an 1 3. b. Yes. an 5 2n for any integer n, including n 1 1. Developing Skills 3. 1, 2, 3, 4, 5 4. 6, 7, 8, 9, 10 5. 2, 4, 6, 8, 10 6. 1, 12, 13, 14, 15 7. 21, 1, 32, 2, 52 8. 19, 18, 17, 16, 15 9. 3, 9, 27, 81, 243 10. 1, 4, 9, 16, 25 11. 5, 7, 9, 11, 13 12. 1, 3, 5, 7, 9 13. 21, 23, 34, 45, 56 14. 3, 2, 53, 32, 75 15. 17. 18. 21, 22, 23, 24, 25 20 4, 16 3, 3 6-2 Arithmetic Sequences (pages 256–257) Writing About Mathematics 1. Virginia’s solution works, but it is not a better method. As the value of n increases, her method becomes more and more time-consuming. 2. No. Pedro’s method yields an arithmetic sequence of six terms, not five. Developing Skills 3. Yes, d 5 3 4. Yes, d 5 2i 5. No 6. Yes, d 5 25 7. No 8. Yes, d 5 0.25 9. a. d 5 3 10. a. d 5 5 b. 24 b. 57 11. a. d 5 22 12. a. d 5 12 16. 9, 6, 3, 0, 23 4 8 3, 3, 1 2 1 19. a. b. 21. a. b. 23. a. i, 1 1 i, 32 1 i, 2 1 i, 52 1 i an 5 2n 20. a. an 5 3n 18 b. 27 an 5 3n 2 2 22. a. an 5 3n 25 b. 39 5 19,683 24 an 5 2n 24. a. an 5 2n 1 5 b. 24 29 < 0.047 25. a. an 5 12i 2 2ni b. 26i n 27. a. an 5 n 1 1 9 b. 10 29. a. an 5 n ? (21)n11 b. 9 31. 5, 6, 7, 8, 9 33. 1, 3, 7, 15, 31 35. 20, 16, 12, 8, 4 37. 108, 36, 12, 4, 43 d. 2n 2 1 b. 0 b. 72 13. a. d 5 22 14. a. d 5 0.1 b. 219 b. 4.0 15. 12, 18, 24, 30, 36, 42 16. 120, 115, 110, 105, 100, 95, 90, 85, 80 17. 6, 9, 12, 15 18. 73, 11 3 19. an11 5 an 2 3 Applying Skills 20. $6,000, $5,600, $5,200, $4,800, . . . The amount owed each month has a common difference, 2400. 21. Week 9 a. 60 5 20 1 (n 2 1)5 b. 20, 25, 30, . . . , 60 c. Using a formula is more efficient for long sequences. 22. a. Choose any linear function and set up a chart, showing that for each integer value of x, y increases by a fixed amount. b. a1 5 b, d 5 m 23. a. 40 b. 154 b. 23 1 26. a. an 5 n 1 1 1 b. 10 28. a. an 5 n2 1 1 b. 82 30. a. an 5 !n b. 3 32. 1, 3, 9, 27, 81 34. 22, 4, 28, 16, 232 36. 4, 5, 7, 10, 14 38. 4, 10, 25, 62.5, 156.25 39. 12, 2, 12, 2, 12 Applying Skills 40. a. 30, 35, 40, 45, 50, 55, 60 b. an11 5 an 1 5, a1 5 30 41. a. 4, 6, 8, 10, 12, 14, 16 b. an11 5 an 1 2, a1 5 4 42. a. 180, 178, 176, 174, 172, 170, 168, 166, 164 b. an11 5 an 2 2, a1 5 180 43. a. Jan 1, Jan 8, Jan 15, Jan 22, Jan 29 b. an11 5 an 1 7, a1 5 1 44. a. $400, $440, $484, $532.40, $585.64, $644.20, $708.62 b. an11 5 1.1an, a1 5 $400 6-3 Sigma Notation (pages 260–261) Writing About Mathematics 1. Yes. The first and last terms of the series have been decreased by 2, and then re-increased by 2 in the expression evaluated by sigma. 299 14580AK02.pgs 3/26/09 12:05 PM Page 300 2. k1 is undefined for k 5 0. Developing Skills 3. a. 3 1 6 1 9 1 12 1 15 1 18 1 21 1 24 1 27 1 30 b. 165 4. a. 0 1 2 1 4 1 6 1 8 b. 20 5. a. 1 1 4 1 9 1 16 b. 30 6. a. 1 1 8 1 27 1 64 1 125 1 216 b. 441 7. a. 95 1 90 1 85 1 80 1 75 1 70 1 65 1 60 1 55 1 50 b. 725 8. a. 12 1 15 1 18 1 21 1 24 1 27 b. 117 9. a. (4 1 2i) 1 (9 1 2i) 1 (16 1 2i) 1 (25 1 2i) b. 54 1 8i 10. a. 21 1 2 2 3 1 4 2 5 1 6 2 7 1 8 2 9 1 10 b. 5 11. a. 14 1 17 1 20 1 23 1 26 1 29 1 32 1 35 1 38 1 41 1 44 b. 319 12. a. 2i 2 2i 2 3i 2 4i 2 5i 2 6i 2 7i 2 8i 2 9i 2 10i b. 255i 13. a. 27 2 11 2 15 b. 233 14. a. 0 1 4 2 64 1 1,296 2 32,768 1 1,000,000 b. 968,468 7 30. a. an 5 45 1 (n2 1) ? 15 5 15n 1 30 5 n51 6-4 Arithmetic Series (pages 264–265) Writing About Mathematics 1. Yes. 1,200 5 n2 (80) , yielding n 5 30. This is true for any arithmetic series with 30 terms such that a1 1 an 5 80. 2. No. The difference between terms is not constant. Developing Skills 3. 42 4. 210 5. 245 6. 60i 7. 7 8. 120!2 10 n51 5 6 n51 n50 5 5 n51 9 n51 14 5 23. a (21) n51 ` n n51 n51 15 n51 n51 b. 165 2 10 10 14. a. a 22(n 2 1) 5 a 22n 1 2 n51 n51 b. 290 12 15. a. a 13n 1 a n(n 1 1) b. 26 n51 20 n51 20 16. a. a f100 2 5(n 2 1)g 5 a f25n 1 105g 26. a A 3nn B ` 25. a n2 15 13. a. a f2 1 12 (n 2 1)g 5 a ( 12n 1 32) 6 24. n51 b. 322 n51 A 3nn B 14 12. a. a f10 1 2(n 2 1)g 5 a (2n 1 8) 1 22. a n! n51 n51 b. 60 5 n 21. a n 1 1 5 11. a. a f24 2 6(n 2 1)g 5 a (26n 1 30) 20. a n12 1 2 n51 19. a 3n n51 b. 72 19 10 6 10. a. a f24 2 4.8(n 2 1)g 5 a (24.8n 1 28.8) 18. a (100 2 5n) n51 n51 b. 210 n51 17. a nn 10 9. a. a f3 1 4(n 2 1)g 5 a (4n 2 1) 16. a (5n 2 4) n51 n51 31. (1) 54.50 (2) 12.74 (3) 0.67 8 15. a (2n 1 1) 5 b. a f45 1 15(n 2 1)g 5 a (15n 1 30) n51 n51 b. 1,050 1 10 35 n51 n51 b. 387.5 12 12 18. a. a f7 1 2(n 2 1)g 5 a (2n 1 5) n51 n51 b. 216 19. a. 2 1 4 1 6 1 c 1 20 b. 110 35 b. a f20 1 3(n 2 1)g 5 a (3n 1 17) n51 10 17. a. a f27.5 1 2.5(n 2 1)g 5 a (2.5n 1 25) Applying Skills 27. ka1 1 ka2 1 c 5 k(a1 1 a2 1 c) 28. (a1 1 b1) 1 (a2 1 b2) 1 (a3 1 b3) 1 c 5 (a1 1 a2 1 a3 1 c) 1 (b1 1 b2 1 b3 1 c) 29. a. an 5 20 1 (n 2 1) ? 3 5 3n 1 17 n51 300 14580AK02.pgs 3/26/09 12:05 PM Page 301 20. a. 5 1 6 1 7 1 c 1 11 b. 56 21. a. 20 1 18 1 16 1 c 1 2 b. 110 22. a. 100 1 95 1 90 1 c 1 5 b. 1,050 23. a. 22 2 4 2 6 2 c 2 50 b. 2650 24. a. 1 1 3 1 5 1 c 1 19 b. 100 Applying Skills 25. 45 26. a. 11 days b. 22 miles 27. $120 28. 2,135 seats 29. 375 minutes 30. $129,000 31. $10,350 37. 38. 39. 40. 41. 6-6 Geometric Series (pages 272–273) Writing About Mathematics 1. Yes. an 5 a1rn21, so anr 5 a1rn. 2. Probably not. This method becomes especially cumbersome with large values of n. Developing Skills 3. 4,095 4. 354,292 5. 536,870,911.5 6. 1,111,110 10 59,048 7. 409.5 8. 32 ?2391 5 39,366 < 1.49997 6-5 Geometric Sequences (pages 269–270) Writing About Mathematics 1. Answers will vary. This method works fine for small sequences, but is inefficient for large values of n. 2. Yes. There are three geometric means between 8 and 32. Developing Skills 3. Yes, r 5 2 4. Yes, r 5 5 5. No, arithmetic 6. Yes, r 5 4 7. Yes, r 5 23 8. Yes, r 5 13 9. 11. 13. 15. Yes, r 5 13 Yes, r 5 210 Yes, r 5 22 1; 6; 36; 216; 1,296 17. 2, 6, 18, 54, 162 19. 21. 22. 23. 24. 25. 26. 27. 9. 11. 13. 15. 10. 2,441,406 12. 6,554 39,364 or 19,684 14. 1,023 a. 3 1 6 1 12 1 24 1 48 1 96 b. 189 1 16. a. 1 1 13 1 19 1 c 1 243 1,275 63 64 364 b. 243 < 1.4979 No, arithmetic Yes, r 5 0.1 Yes, r 5 a 40, 20, 10, 5, 52 17. a. 10 1 5 1 52 1 c 1 56 b. 315 16 5 19.6875 18. a. 26 2 24 2 96 2 384 2 1,536 2 6,144 2 24,576 2 98,304 2 393,216 b. 524,286 19. a. 1 2 2 1 4 2 8 1 16 2 32 b. 221 32 20. a. 1 1 23 1 49 1 c 1 243 18. 14, 21 2 , 1, 22, 4 20. 10, 30, 90, 270, 810 1, !2, 2, 2!2, 4 21, 4, 216, 64, 2256 100, 10, 1, 0.1, 0.01 or 100, 210, 1, 20.1, 0.01 1, 4, 16, 64, 256 or 1, 24, 16, 264, 256 1, !2, 2, 2!2, 4 or 1, 2!2, 2, 22 !2, 4 1, 22, 4, 28, 16 81, 27, 9, 3, 1 or 81, 227, 9, 23, 1 128 28. 0.00032 or A 15 B 29. 30. 31. 32. 10. 12. 14. 16. $3,150, $3,307.50, $3,472.88, $3,646.52 5,000; 4,900; 4,802; 4,706; 4,611; 4,520; 4,429; 4,341 55, 61, 67, 73, 81 $16,000, $12,800, $10,240, $8,192 $42,500, $36,125, $30,706.25, $26,100.31, $22,185.27, $18,857.48 b. 665 243 < 2.7366 21. a. 100 1 50 1 25 1 c 1 100 64 b. 5 3,175 16 5 198.4375 22. a. 281 1 27 1 29 1 c 1 21 9 256 2,187 6256!2 b. 2547 9 5 260.7 23. 13 1 13 !3 8 7 24. 1 2 6251 < 1,038.66 1 2 6257 25. 1,023 26. a. 400(1.05)1 5 $420 b. Yes, r 5 1.05 c. $2,856.80 27. 1789 feet 28. 20 days 1 3 33. 15, 37.5 64 16 64 34. 4, 16 3 , 9 or 24, 3 , 2 9 35. 24 !2, 144, 432 !2 or 224 !2, 144, 2432!2 Applying Skills 36. $1,000, $1,060, $1,123.60, $1,191.02, $1,262.48, $1,338.23, $1,418.52, $1,503.63, $1,593.85, $1,689.48 301 14580AK02.pgs 3/26/09 12:05 PM Page 302 6-7 Infinite Series (page 278) 3. a. b. c. d. 4. a. b. d. 5. a. b. d. 6. a. b. c. d. Writing About Mathematics a 1. S 5 1 2 r 5 1 1 2 1c c 5c2 1 2. No. The calculator’s value of e is only an approximation. e is an irrational number. Developing Skills 3. a. 1 1 a A 13 B ` n n51 b. Finite limit: 32 4. a. 2 1 a 2 A 14 B ` n n51 b. Finite limit: 83 8 5. a. a 2n 8. a (3n 1 2) n51 6 n51 6. a. 5 1 a 5 A 15 B ` n n51 ` n51 b. Decreases without limit n n51 b. Finite limit: 12 ` 9. a. a (n 11 1)! n51 b. Finite limit: (e 2 2) ` n(n 1 1) a 2 n51 12. 13 13. 49 14. 12 99 15. 24 99 16. 126 999 2 2,048 an11 5 an 1 (6 1 2n), a1 5 12 1, 7, 31, 127, 511 6, 11, 16, 21, 26, 31 5, 25, 125 or 25, 25, 2125 12 25. 26. 27. 28. 3 1 9 1 27 1 81 5 120 12 1 9 1 6 1 3 1 0 2 3 2 6 5 21 60.26 !2 2 ` 11. 10 9 14. 151 16. 220 9 18. 58 < 7,629.3945 19. 20. 21. 22. 23. 24. 29. a. a 3 ? A 12 B b. Increases without limit 12. a 21n n51 13. 6, 11, 16, 21, 26 15. 71 125 625 17. 2, 5, 25 2, 4 , 8 7. a. 5 1 a (5 2 4n) n21 n51 30. 17. 1 # n , 25 31. Review Exercises (pages 280–281) 32. an+1 5 an 1 4, a1 5 1 Arithmetic an 5 1 1 4(n 2 1) 5 4n 2 3 37 a an 1 1 5 3n , a1 5 3 Geometric c. an 5 3 A 13 B 1 d. 6,561 n51 ` 11. a (21) n21 ? n b. Finite limit: 25 4 ` 10. a (2n 2 1) 2 n51 7 n51 8. a. 6 1 a 6 A 12 B n50 6 9. a (2n 1 2) b. Increases without limit 1. a. b. c. d. 2. a. b. 6 (n)(n 1 1) 2 7. a ` 10. a. an11 5 an 2 1, a1 5 12 Arithmetic an 5 12 2 1(n 2 1) 5 2n 1 14 3 an11 5 an 1 n 1 1, a1 5 1 Neither 55 an11 5 an 1 i(2n), a1 5 i Neither 1,023i an11 5 23an, a1 5 2 Geometric an 5 2(23) n 2 1 239,366 33. b. a. b. a. b. a. b. a. b. Finite limit: 6 8 cans 108 cans $24,500 $222,500 $52,637.27 $368,569.05 an 5 an–1 1 n 2 1 0, 1, 3, 6, 10, 15, 21, 28, 36, 45 c. an 5 n21 302 n(n 2 1) 2 14580AK02.pgs 3/26/09 12:05 PM Page 303 Exploration (page 282) Part IV In 1–4, part a, answers will be graphs. 1. b. Diverges 2. b. Oscillates 3. b. Converges 4. b. Converges 1 15. a. y 5 x 2 3 b. y Cumulative Review (pages 283–285) Part I 1. 4 4. 1 7. 1 10. 4 Part II 2. 2 5. 3 8. 3 x 3. 1 6. 3 9. 2 c. 11. Answer: 49 A x 1 32 B 2 A 212, 212 B Set both equations equal to each other: x2 1 3x 1 94 5 94 12. 7 2 !x 1 2 5 4 2!x 1 2 5 23 x1259 x57 Part III 13. 211 O x 2 1 3 5 94 5 3x 1 1 x 2 1 5 9x 1 3 28x 5 4 x 5 212 Substitute this value of x into either equation to find the y-coordinate. 16. a. an11 5 10an, a1 5 3 (2 1 i) 2 i 3 1 4i 5 i 5 3 1i 4i ? ii 2 4 5 3i 21 5 b. a 3(10) n 2 1 n51 c. 33,333 5 4 2 3i 14. 2 1 0 1 2 3 4 5 Chapter 7. Exponential Functions 7-1 Laws of Exponents (pages 288–289) Applying Skills 27. 9 29. y 5 x3 31. $608.33 33. 15 years Writing About Mathematics 1. No, they do not share a common base or common exponent. (2)3(5)2 5 (8)(25) 5 200. 105 5 10,000. 2. Yes, this is true via the commutative property. Developing Skills 3. x7 4. y6 5. x4 6. y3 7. x10 8. 8y12 9. 106 10. 228 11. x6y3 12. x2y7 13. 29x6 14. 9x6 5 15. x y 16. x 17. x8y10 18. 64x10 19. 16 20. x5y5 21. xy2 22. x2y2 24. 32a5b 25. 4abc4 28. 3 30. x 5 25y2 32. $3,909.35 7-2 Zero and Negative Exponents (pages 292–293) Writing About Mathematics 1. No. a0 1 a0 5 2a0 5 2(1) 5 2 2. Yes. a0 1 a0 5 1 1 1 5 2 and 2a0 5 2. Developing Skills y7z2 23. x2 26. b 303 3. 15 1 4. 16 1 5. 36 6. 2 7. 125 8. 32 14580AK03.pgs 9. 12. 15. 18. 21. 24. 3/27/09 16 1 1 1 1 10:53 AM 10. 13. 16. 19. 22. 25. 730 243 1 400 1 4 1 3 1 21,259,712 A 43 B 11. 14. 17. 20. 23. 26. 1 4 1 (26) 28 15 30. !3 74.8309 3 64 34. 56 35. 37. y15 38. 40. 25 y8 41. 43. 12 1 44. 24x 2 45. 1 46. 4x 2 47. x3 48. y7 49. 3a5 52. 2x12 50. 6a4 53. a4 51. 9a3x2 54. y22 55. z25 15 12 56. b a15c 8m9 57. 27n 6 58. 2ab41x3y4 61. 232b4 xa2 64. 67. 70. 73. 76. y22 6b21 2a23 5xy210 218 59. y2 33. 36. 39. 42. 45. 13 46. 5a2 54. 62. 63. x4 1 1 x5 65. 68. 71. 74. 77. ab22 a3 4x5 5x23y23 66. 69. 72. 75. 78. y24 3x24 3a 25a6b24 a58 b53 3a 2 A 4b B 1 1 1 (xy) 5 z 7 1 4 8 22. 58. !3 3 7 5 6 3 70. " xyz 5 232x 22x 72. # 5 y4 5 y4 5 " 2 10 6 5 3 2 4 2 74. 64a 729 2 x3 y3 75. 8 4 3 3 2 3 3xb 2 3 77. 2a z3 78. x3y2 79. 11 a6 1 1 2x 1 3 5 3 3 ? A273 ? 33 B 2 83. a. A !27 ? !3 B 2 5 1 1 1 3 8 ? " 3 5 A31 1 3 B 2 1 8 8 33 5 33 ✔ b. ? Q !3 R 5 Q 31 R !9 1 2 3 3 92 !3 3 ? 32 Q !3 R 5 23 !3 3 A 1 2 23. 9 26. 8 304 1 3 ? 5 !3 B (3 ) 2 3 32 2 3 ? 1 1 5 3 3 ✔ 32 32 7 485 5 215 7 14 7 14 2 6 x15 y15 315 x15 y15 81. 5 6 82. 22xy4 10 6 2 343 1,000 y3 4 4 3 x3 b3 2 3 3x 113 y3 1 2 2 5 7 5 5 2 2 76. 2a5b 3 ? A! 81 B 2 5 A(33) 3 ? 33 B 2 5. 8. 11. 14. 17. 20. 2 6 8a b " 216a b c 73. # 3c 27c4 5 1 5 3 2 y2 68. 5b2 !2a 3 5a "10a 71. # 4 5 2 5 3 3 7 6 66. x " x 4 69. 2ab" ab2 4 5 ! a 25. 49 y 64. 36!6 2 5 2 56. w x 5 w x1 7 32 1 32 !5 25 0 1 21. 1,000 24. 5 1 15 10 2 5 10 3 55. 2ab 2 67. 5x!y n 1. A ! a B 0 5 an 5 a0 5 1 1 3 1 25 53. 644x4 5 22x4 63. 12!12 65. Writing About Mathematics 3 5 6 240 81 52. 3a 62. 5!5 3 1 3 50. 3ba 3 61. ! 9 80. 1 1 49. 23 (2) a3b 4 7-3 Fractional Exponents (pages 296–298) 1 1 47. 7x 1 2 57. (a ? b ) 5 a ? b 3 59. !5 60. ! 6 1 79. 3 3 100 2. Aa2 B 2 5 a2 ?2 5 a4 Developing Skills 3. 2 4. 6. 2 7. 9. 2 10. 12. 15 13. 15. 32 16. 18. 16 19. 1 41. 153 4 44. 95 1 1 4 60. 9b uv3 7 4 9 1 48. 8a b3 24 5 1 40. 123 3 43. 25 3 2 12 5 38. 72 39. 62 1 42. 34 51. 29. 32. 35. 9 4 37. 1 5 6128 1 or !2 2 !2 1 a6 7 a4 1 81a4 1 281a 4 1 x4 2 x2 1 4x2 x9 28. 2 31. 2 1 34. 381 52 32. xy 64 9 112 36. 1.196 1 1 32 31. 27. 30. 33. 1 21 22 9 28. 10 1016 1 5 1,000.000001 27. 14 29. Page 304 1 22x2 !y y 5 14580AK03.pgs 3/27/09 10:53 AM Page 305 7-4 Exponential Functions and Their Graphs (pages 302–303) 10. a. Writing About Mathematics 1. Any non-zero number raised to the zero power is 1. 2. One raised to any power is 1, thus y 5 1 for all values of x. Developing Skills In 3–6, parts a and b, answers will be graphs. x 3. c. y 5 42x or y 5 A 14 B 4. c. y 5 32x or y 5 A 13 B 2x x 5. c. A 72 B or A 27 B 6. c. x ex –2 0.135 –1 0.368 0 1 1 2.718 2 7.389 3 20.086 b. y x 7. a. A 34 B 2x x or A 43 B 4 y 3 4 2 3 1 2 x 21 c. 1.6 Applying Skills 11. a. y 1 x O b. 3.1 8. a–b. 450 400 350 300 250 200 150 100 50 2 1 c. 4.4 y 1 O 1 O x c. y 5 22x y 1 O 1 x O 1 2 3 4 5 6 7 8 9 10 11 12 1314 b. In 2010, 338,880,723. In 2020, 386,313,106. 9. a–b. 1 x c. y 5 21.2x 305 14580AK03.pgs 12. a. 3/27/09 9 8 7 6 5 4 3 2 1 10:53 AM Page 306 Developing Skills 3. 32 6. 72 y 9. 12. 15. 18. 21. x 10 ,0 20 00 ,00 30 0 ,0 40 00 ,0 50 00 ,0 60 00 ,00 70 0 ,0 80 00 ,0 90 00 ,00 0 O b. In 10 years, 9.997 grams. In 100 years, 9.971 grams. c. Answers will vary: 79,951 years 13. a. Graph b. 2 points: (2, 16) and (4, 64) c. y 5 4x A 12 B 3 or 223 4. 33 7. 103 10. (0.5)3 4 22 212 13. 16. 19. 22. A 16 B 5. 52 8. 25 3 or 623 11. (0.1)3 (0.9)2 3 2 2 14. 17. 20. 23. (0.4)2 21 22 3 24. 21 25. 12 26. 22 27. 30. 33. 36. 28. 31. 34. 37. 29. 32. 35. 38. 3 3 1 0 3 2 1 5 23 6 22 3 2 62 7-7 Applications of Exponential Functions (pages 312–313) Writing About Mathematics 1. 100% 5 1; thus, A 5 A0(1 1 1)n 5 A0(2)n. 2. Daily interest earns interest on earned interest, not just the principal. Developing Skills 3. 7.39 4. 4.48 5. 0.37 6. 2.72 7. 0.23 8. 2.72 9. 2,980.96 10. 168.50 11. 51.01 12. 344.60 13. r 5 100% 14. t 5 3 15. 577.21% 16. 236.11% Applying Skills 17. a. $1,060, $1,123.60, $1,191.02, $1,262.48, $1,338.23 b. $1,061.68, $1,127.16, $1,196.68, $1,270.49, $1,348.85 c. Sue d. Joe 5 6%, Sue 5 6.168% 18. a. $10,129.08 b. $10,272.17 19. $1,508,661.82 20. 17.22 g 21. 31,529 22. $369,452.80 23. 30.23 g 24. 4,000 25. a. A 5 A0ert; medicine decreases continually. b. Continuous 5 69.99 mg Periodic 5 66.16 mg 7-5 Solving Equations Involving Exponents (page 305) Writing About Mathematics 1. Yes. Squaring both sides eliminates the fractional exponent. 2. No. a22 5 a12 , but 36 does not equal its inverse. Developing Skills 3. 64 4. 32 5. 243 6. 4 7. 613 8. 2 9. 16 10. 81 11. 16 12. 9 13. 27 14. 72 15. 3 16. 5 17. 81 18. 0.35 19. 14.70 20. 1.24 21. 2.03 22. 2.20 23. 0.54 1 2 23 3 24. 5 10 x 1 x23 5 10 1 Ax23 B 23 5 (10) 23 1 x 5 1,000 Applying Skills 25. If the area of one face is B, then the length of one side of the cube is !B. Therefore, the volume of 3 the cube is A !BB 3 or B2 . 2 26. B 5 V3 Review Exercises (pages 315–316) 7-6 Solving Exponential Equations (pages 307–308) Writing About Mathematics 1. a 5 0. Anything to the zero power is 1. 2. There is no common base. 1. 1 2. 12 3. 21 4. 5 5. 64 6. 500 7. 36 1 8. 36 9. 3 1 10 11. 25 1 12. 64 10. 13. 2 306 14. 1 28 1 15. 1616 14580AK03.pgs 3/27/09 10:53 AM Page 307 16. 10,000 17. 59,049x2 1 xb 22. 32 y3 20. 6 18. ac6 or 1 19. 3 2 5 5 6yz2 x 21. A ac B 6 12. 3 1 5x2 x2 2 92x 1 5 5 0 23. bA64 a b B or 2 a b 1 12 1 6 4 3 5 6 1 2 4 3 7 12 2 3 1 2 7 12 5 3 or 2x2 2 9x 1 10 5 0 1 2 24. 32 x y or 2 x y 25. 4y2 !2y 13. 4 28. ab"a2b2c 4 27. Part III 4 26. 4x !y A !a 1 2B 3 8 29. a–b. y 1 21 O c. rotation about the y-axis 9 4 3 2 30. 33. 1 6 1 2 31. 64 34. 21 32. 35. 36. 39. 42. 45. 22 21 $520.30 $6,553.60 37. 40. 43. 46. 38. 4 41. 21 44. 4.7% 2 1 2 9.05 mg (0.5, 0.346) Part IV 15. Answer: 7 2 2!97 , x , 7 1 2!97 Use the quadratic formula to find the roots of the corresponding equation: x5 5 Exploration (page 316) 8 9 4 3 5 8 5 12 5 32 1 4 b. 2a154? 2a21 1 16 5 22a14 1 16 4(4a 1 1) 4(4a 1 1) 5 4a12 1 42 5 42 (4a 1 1) 5 14 Cumulative Review (pages 316–318) Part I 1. 2 4. 2 7. 2 10. 4 2. 2 5. 1 8. 4 2(27) 6 " (27) 2 2 4(1)(212) 2(1) 7 6 !49 1 48 2 7 6 !97 < 21.42, 8.42 2 Test a number from each interval formed by the roots to find the solution. 16. a. (x 2 2)2 1 y2 5 16 b. x2 1 y2 2 4x 2 12 5 0 c. y 5 x 1 2 d. Answer: (2, 4), (22, 0) Substitute y 5 x 1 2 into the equation of the circle: 4(4a 1 1) a11 3 1 2i 1 1 2i 1 2 2i ? 1 1 2i 2 4 5 3 1 4i 5 7 5 21 5 1 5i 14. 1 1 27x11 5 82 27x11 5 81 33x13 5 34 33x 5 31 x 5 13 x a a22 3a (1 2 322) a. 33a21213 3a 5 3a (321 1 1) 5 (x 2 2) A x 2 52 B 5 0 x2 2 52x 2 2x 1 5 5 0 x2 1 (x 1 2) 2 2 4x 2 12 5 0 3. 3 6. 2 9. 1 x2 1 x2 1 4x 1 4 2 4x 2 12 5 0 2x2 2 8 5 0 2x2 5 8 x 5 62 Part II 11. The common difference d is 2. The first term a1 is 1. Substitute x 5 62 into the equation of the line to find the y-coordinates. an 5 1 1 (n 2 1)2 5 2n 2 1 307 14580AK03.pgs 3/27/09 10:53 AM Page 308 Chapter 8. Logarithmic Functions 8-1 Inverse of an Exponential Function (page 323) 7. a. f21 (x) 5 log0.5 x b. y Writing About Mathematics 1. Yes. The point (0, 1) is on the graph of any exponential function y 5 bx. Therefore, since y 5 logb x is the inverse of the exponential function y 5 bx, (1, 0) is always on its graph. 2. Yes. x 5 b2y, thus 2y 5 logb x, y 5 12 logb x. Developing Skills 3. a. f21(x) 5 log3 x b. y 1O 21 1 O 1 8. a. f21 (x) 5 log!2 x b. y 1 x 9. a. f21 (x) 5 log13 x b. y 1 O 1 x 5. a. f21(x) 5 log1.5 x b. y 1 O x 10. a. f21(x) 5 log2 (2x) b. x y 1O 1 21 6. a. f21(x) 5 log2.5 x b. y 1O x O 21 4. a. f21(x) 5 log5 x b. y 1O 21 x 11. 13. 15. 17. 19. 21. x 1 308 y 5 log6 x y 5 log8 x y 5 log0.2 x y 5 log121 x y 5 5x y 5 8x x 12. 14. 16. 18. 20. 22. y 5 log10 x y 5 log0.1 x y 5 log14 x y 5 2x y 5 10x y 5 (0.1)x or y 5 10–x 14580AK03.pgs 3/27/09 10:53 AM Page 309 8-3 Logarithmic Relationships (pages 331–332) Applying Skills 23. a. (0, 1), (1, 3), (2, 9), (3, 27) b. (1, 0), (3, 1), (9, 2), (27, 3) 24. a. (0, 1), (1, 1.05), (2, 1.10), (3, 1.16) b. (1, 0), (1.05, 1) (1.10, 2), (1.16, 3) Writing About Mathematics 1. loga an 5 nloga a 5 n ? 1 5 n by the logarithm of a power rule and the logarithm of the base rule. 2. No. For example, log10 10 ? 10 5 2 and (log10 10) ? (log10 10) 5 1. Developing Skills 3. log3 (27 3 81) 5 log3 27 1 log3 81 5 3 1 4 5 7 27 3 81 5 37 5 2,187 4. log3 (243 3 27) 5 log3 243 1 log3 27 5 5 1 3 5 8 243 3 27 5 38 5 6,561 5. log3 (19,683 4 729) 5 log3 19,683 2 log3 729 592653 19,683 4 729 5 33 5 27 6. log3 (6,561 4 27) 5 log3 6,5612 log3 27 5 8 2 3 5 5 6,561 4 27 5 35 5 243 7. log3 94 5 4 log3 9 5 4 3 2 5 8 94 5 38 5 6,561 8. log3 2432 5 2 log3 243 5 2 3 5 5 10 2432 5 310 5 59,049 9. 2 log3 81 1 log3 9 5 2 3 4 1 2 5 10 812 3 9 5 310 5 59,049 1 10. 2 log3 6,561 2 log3 729 5 4 2 6 5 22 8-2 Logarithmic Form of an Exponential Equation (pages 326–327) Writing About Mathematics 1. b2a 5 b1a , so if ba 5 c, then b2a 5 1c . 2. If ba 5 c, then b2a 5 c2. Developing Skills 3. log2 16 5 4 5. log8 64 5 2 7. log6 216 5 3 9. log5 0.008 5 23 4. 6. 8. 10. 11. log7 17 5 21 13. log625 125 5 15. 102 5 100 17. 42 5 16 19. 35 5 243 21. 10–3 5 0.001 23. 5–2 5 0.04 3 25. 492 5 343 27. 1 30. 12 33. –2 36. 2 39. 4 42. 3 45. 2 48. 2 51. 90 54. 2 28. 31. 34. 37. 40. 43. 46. 49. 52. 55. 12. log64 4 5 13 14. log100 0.001 5 232 16. 53 5 125 18. 27 5 128 20. 70 5 1 22. 100–1 5 0.01 1 24. 83 5 2 2 26. 3225 5 0.25 29. 3 32. 28 35. 6 38. 216 3 41. 16 44. 5 47. 2 50. –4 53. 8 56. 59 57. 60. 63. 58. 5 59. 4 61. 3 62. 2 1 64. 390,625 5 0.00000256 1,000 2 1 25 1 10 3 4 5 –1 26 –3 4 3 2 23 8 16 15 log5 125 5 3 log12 1 5 0 log10 0.1 5 21 log4 0.0625 5 22 67. 12 68. 100 66. 2 !2 69. 4 70. 21 71. 23 72. 35 1 73. 10 65. !6,561 4 729 5 322 5 19 11. 14 (log3 243 1 log3 2,187) 5 14 (5 1 7) 5 3 4 ! 243 3 2,187 5 33 5 27 12. 12 (log3 19,683 2 log3 2,187) 5 12 (9 2 7) 5 1 !19,683 4 2,187 5 31 5 3 13. 3 log3 81 2 12 log3 729 5 12 2 3 5 9 813 4 !729 5 39 5 19,683 14. log3 27 1 13 (log3 729 2 log3 19,683) 5 3 1 13 (6 2 9) 5 2 3 729 5 32 5 9 27 3 #19,683 16. 15. a. log3 9 b. 2 17. a. log3 13 18. b. 21 19. a. log3 27 20. b. 3 74. 4 a. b. a. b. a. b. log3 2,187 7 log3 27 3 log3 3 1 21. a. 4(log3 9 2 log3 27) 22. a. 12 (log3 3 1 log3 243) b. 24 b. 3 23. a. log4 16 b. 2 24. loge 10x 25. log2 ab Applying Skills 75. t 5 log1.06 A 76. n 5 log0.97 R 77. a. loge A or ln A b. t 5 2,500 loge A or t 5 2,500 ln A 26. log2 (x 1 2)4 309 y 27. log10 (y 2 1) 2 14580AK03.pgs 3/27/09 10:53 AM Page 310 x ? y2 28. loge z2 29. log3 x3 30. log2 2 1 log2 a 1 log2 b 5 1 1 log2 a 1 log2 b 31. log3 10 2 log3 x 32. 25 log5 a 33. 2 log10 (x 1 1) 34. 6 log4 x 2 5 log4 y 35. 12 loge x 36. A 1 B 37. 2A 1 B 39. A 1 3B 38. 3(A 1 3) 40. A 2 B 41. 2A 2 3B 42. 12 (A 1 B) 43. A 1 12B B 45. A 2 2 44. 12A 2 3B 57. Applying Skills 58. a. K 1 48. 32 49. 32 50. 32 53. 10 51. 25 52. 2 0 t 8-4 Common Logarithms (pages 335–336) 70 60 50 40 30 20 10 3 4 5 15.403 24.414 30.807 35.765 20 51.169 66.572 t b. 2 10 K 46. 32A 47. 14B 82 ? (x2 2 4) 32(x2 2 4) 5 log 3 6 1 1 1 2 log x 1 2 log y 2 2 log z 56. log 30 75.582 t K O 5 10 15 20 25 30 Writing About Mathematics 1. log 80 5 log (10 3 8) 5 log 10 1 log 8 5 1 1 log 8 2. 10 must be raised to a negative power to yield values less than 1. Developing Skills 3. 0.57 4. 0.93 5. 1.68 6. 1.75 7. 2.75 8. 3.75 9. 20.47 10. 21.12 11. 0 12. 1 13. 2 14. 21 15. 3.01 16. 1.90 17. 22.70 18. 1.43 19. 0.60 20. 79.59 21. 2.58 22. 1.57 23. 21.49 24. 3.7905 25. 6.7562 26. 24.1324 27. 60.1174 28. 159.5879 29. 364.6700 30. 66,069.3448 31. 0.2902 32. 0.8764 33. 0.0701 34. 0.0010 35. 0.0001 36. x 1 y 37. 2x 38. 2y 39. 2x 1 y 40. x 1 2y 41. 3x 42. 2x 43. 2y 44. 22y 45. 2x 2 1 46. x 2 y 47. 2(x 2 y) 48. 2c 49. 1 1 c 50. 2 1 c 51. c 2 1 52. 2 2 c 53. 2c 2 1 54. 2c 2 2 55. 12c c. Double in the 15th year, triple in the 24th year 59. a. 7.40 b. 2.19 c. 4.40 8-5 Natural Logarithms (pages 338–339) Writing About Mathematics 1. The bases of the logarithms do not affect the answer. If loga x 5 y and logb x 5 z, then ay 5 bz 5 x. 2. a 5 1. logb 1 5 0 for any positive b 1. Developing Skills 3. 1.32 4. 2.15 5. 3.87 6. 4.03 7. 6.33 8. 8.63 9. 21.07 10. 22.58 11. 0 12. 1 13. 2 14. 21 15. 20.69 16. 3.56 17. 0.55 18. 13.82 19. 0.39 20. 0.35 21. 1.7837 22. 2.2926 23. 3.9852 24. 5.9239 25. 9.0521 26. 12.9604 27. 123.9651 28. 0.5843 29. 0.9443 30. 0.3152 31. 0.3679 32. 0.1353 33. x 1 y 34. 2y 35. 2x 36. 2x 1 y 37. 3x 1 y 38. 2x 1 2y 39. 2y 40. 2x 41. 2x 2 y 42. 22(x 1 y) 43. x 2 y 44. 2(x 2 y) 45. 2c 46. 3c 47. 2c 48. 2c 49. 22c 50. 22c 51. 12c 52. 21c 53. 3.555 54. 5.380 55. 0.693 56. { } 3 57. ln !xy? z 58. 2 ln e 1 ln x 1 12 ln y 2 ln z 5 2 1 ln x 1 12 ln y 2 ln z 310 14580AK03.pgs 3/27/09 10:53 AM Page 311 Hands-On Activity:The Change of Base Formula a. 2.11 b. 4.09 c. 1.16 d. 1.84 e. 20.73 f. 22.58 g. 4.95 h. 20.39 14. 16. 18. 20. 22. 24. 26. 28. 8-6 Exponential Equations (pages 343–344) Writing About Mathematics 1. No. You must take the logarithm of each side, not each term. This equation can be solved by first subtracting 6 from each side and then taking the log of both sides. 2. No. The exponent is applied only to 3, not to the entire left side. Developing Skills 3. 2.26 4. 4.17 5. 2.86 6. 1.70 7. 2.23 8. 4.08 9. 6.86 10. 2.38 11. 20.5 12. 20.15 13. 2.14 14. 2.89 Applying Skills 15. 15 years old 16. 5 years 17. 25.5 years 18. 25 years 19. a. 66 minutes 20. a. 20.00251 b. 22 minutes b. 828 days or y 5 log82 x 6. log6 36 5 2 8. log3 !3 5 12 10. log2 14 5 22 15. 17. 19. 21. 23. 25. 27. 29. 56 9 214 22 4 2a 1 2 (a 1 2b) 30. 2(b 2 a) 32. 2a 2 32b 34. 2.5 31. 33. 35. 36. 9 37. 1021 5 0.1 9 212 3 5 2 a1b a 1 2b 1 2b 2 a 1 3 (a 2 b) 1 5 2 1 36 39. !5 4 y 51. A 5 x ? y3 50. A 5 x 2 52. A 5 A xy B 1 3 53. A 5 x ? y3 or A 5 x!y 54. 3.5 57. 3 60. 2,013 55. 3 58. 2 61. 16 Exploration (pages 350–351) Steps 1–8. Review Exercises (pages 348–350) 4. y 5 1 41. a. log4 !4881 b. 22 42. a. log360 (5 3 12 3 6) 5 log360 360 b. 1 16 1 5 log0.5 16 43. a. log0.5 256 b. 4 44. a. log1.5 A 32 3 3 3 12 B 5 log1.5 94 b. 2 45. a. 2 ln 42 2 ln 3 b. 6.38 46. a. 2 ln 14 1 ln 0.625 b. 4.81 47. a. 4 ln 0.25 2 ln 26 1 5 ln 3 b. 23.31 48. A 5 xy 49. A 5 xy Writing About Mathematics 1. No. The left side must first be combined using the rules for logarithms: log x 1 log 12 5 log 12x. Thus, the equation can be solved by writing 12x 5 9. 2. Yes. Taking the logarithm of a number equal to the base is equivalent to 1. Then log x 5 log (10 ? 5). Developing Skills 3. 25 4. 6 5. 1.5 6. 30 7. 126 8. 192 9. 4 10. 9 11. 3 12. 4 13. 5 14. 0.5 15. 500 16. 1 17. {1.38, 3.62} 18. 3.65 log x log 82 13. 42 5 8 72 5 !7 e0 5 1 38. 23 40. 41 8-7 Logarithmic Equations (page 346) 1. a. Graph c. {all real numbers} e. y 5 3x 2. y 5 6x 3 12. 53 5 125 b. {x : x . 0} d. Graph an gn 0 1 1 10 3. y 5 2.5x 1.125 13.3352143 5. log2 8 5 3 1.25 17.7827941 7. log10 0.1 5 21 9. log8 4 5 23 1.5 2 11. 34 5 81 311 31.6227766 100 56. 0.75 59. 1.24 62. 4,500 14580AK03.pgs 3/27/09 10:53 AM Page 312 Part III 53x 13. log 53x 3x log 5 x Step 9. log (13.3352143) < 1.125 Step 10. The mean of 1.25 and 1.5 is 1.375. Thus, log (23.71373706) 1.375. Cumulative Review (pages 351–352) Part 1 1. 4 2. 2 4. 4 5. 4 7. 4 8. 3 10. 3 Part II 11. f(x) 5 4x3 2 x 5 0 x(4x2 2 1) 5 0 x50 4x2 5 1 x 5 612 5 5 5 5 1,000 log 1,000 3 1 log 5 < 1.43 !200 1 !50 1 2!8 5 10!2 1 5!2 1 4!2 5 19!2 Part IV 15. a. 30 1 6 1 1.2 1 0.24 1 c 1 0.000384 b. 37.499904 16. 9.25 5 15e20.000124 t ln 9.25 5 ln 15 2 0.000124 t 14. 3. 4 6. 2 9. 2 15 2 ln 9.25 t 5 ln 20.000124 t < 3,900 years 12. Answer: x2 2 10x 1 34 5 0, 5 6 3i Let a 5 1. r1 1 r2 5 2b 5 10 r1r2 5 c 5 34 Chapter 9.Trigonometric Functions 12. sin 45° 5 cos 45° 5 !2 2 , tan 45° 5 1 9-1 Trigonometry of the Right Triangle (pages 356–357) !3 13. sin 45° 5 12 , cos 45° 5 !3 2 , tan 45° 5 3 Writing About Mathematics 1. They are equal. By definition, if A is an angle on a right triangle, then sin A 5 ha and cos (90 2 A) 5 ha . 2. Yes. Since lengths are positive values and the length of a leg of a right triangle is always smaller than the length of the hypotenuse, sin A is a positive value less than 1. Applying Skills 14. sin 5 45 , cos 5 53 , tan 5 34 15. 0.25 5 5 16. sin 5 13 , cos 5 12 13 , tan 5 12 17. 15 m b. 45 c. 43 5 4. a. 13 b. 12 13 5 c. 12 5. a. 11 61 b. 60 61 11 c. 60 8 6. a. 17 b. 15 17 8 c. 15 8 7. a. 17 b. 15 17 8 c. 15 8. a. !5 5 9. a. !2 3 10. a. 23 b. 2 !5 5 b. !7 3 Writing About Mathematics 1. Yes, 810° 5 90° 1 (2)360°. 2. No, two angles that add to 360 are not necessarily coterminal. For example, 150° and 210°. Developing Skills In 3–7, answers will be graphs. 3. In quadrant I 4. Same as 180° 5. Same as 180° 6. Same as 240° 7. In quadrant II 8. I 9. II 10. III 11. IV 12. IV 13. II 14. I 15. IV 16. I 17. IV 18. 30° 19. 52° 20. 280° 21. 350° 22. 275° 23. 90° c. 21 c. !14 7 c. 2 !5 5 11. The triangles are similar and therefore have the same trig ratios. b. !5 3 19. 56 ft 9-2 Angles and Arcs as Rotations (pages 360–361) Developing Skills 3. a. 35 18. 125 ft 312 14580AK03.pgs 3/27/09 10:53 AM 24. 220° 25. 180° 27. 260° Applying Skills 28. Clockwise 29. 30. a. Clockwise b. 31. 60 32. 33. a. 87 s 34. b. 6 min, 15 sec Page 313 Hands-On Activity 3. Answers will vary: (0.94, 0.34) 5. The values are about the same. 6. Answers will vary: 70° (0.34, 0.94); 100° (20.17, 0.98); 165° (20.97, 0.26); 200° (20.94, 20.34); 250° (20.34, 20.94); 300° (0.50, 20.87); 345° (0.97, 20.26) 26. 0° Counterclockwise 2,340° 12.5 a. 18° per second b. 540° per second c. 3,600° per second In each case, the values of the sine and cosine are approximately equal to the coordinates of P. 9-3 The Unit Circle, Sine, and Cosine (page 366) Hands-On Activity: Finding Sine and Cosine Using Any Point on the Plane Writing About Mathematics 1. Since P is a point on the unit circle, the largest value for either x or y is 1 and the smallest value is 21. 2. No. For example, sin 45° 5 sin 135°. Developing Skills (1) r 5 5, sin u 5 45, cos u 5 35 25 (2) r 5 13, sin u 5 12 13 , cos u 5 13 3. a. 4 5 4. a. 20.8 5. a. 21 6. a. 22 !5 5 7. a. 212 13 8. a. 9. a. 10. a. 11. 13. 15. 17. 19. 21. 7 225 !2 2 40 41 90° 0° (0, 1) (0.2, 1.0) (20.7, 20.8) (21, 0) 23. a. 6 !5 3 b. 3 5 8 (3) r 5 17, sin u 5 215 17 , cos u 5 17 !53 !53 (4) r 5 !53, sin u 5 2753 , cos u 5 2253 !2 (5) r 5 5 !2, sin u 5 710 , cos u 5 !2 10 !2 (6) r 5 5 !2, sin u 5 710 , cos u 5 2!2 10 c. I b. 0.6 c. IV b. !5 5 c. II (7) r 5 !5, sin u 5 225!5, cos u 5 !5 5 c. IV 2!2 (8) r 5 3!2, sin u 5 2!2 2 , cos u 5 2 5 b. 213 c. III 24 25 !2 2 9 241 c. IV b. 2 !3 2 b. b. b. 12. 14. 16. 18. 20. 22. 9-4 The Tangent Function (pages 372–373) Writing About Mathematics 1. a. 45° and 225°. If P is a point on the unit circle and on a 45° angle in standard position, an isosceles right triangle is formed by the x- and y-coordinates of P. Thus, the x- and y-coordinates of P are equal and the sine and cosine of 45° are equal. A similar result holds for 225° by symmetry. sin u b. 45° and 225°. Since cos u 5 tan u, if tan u 5 1, then sin u 5 cos u. sin u 2. cos u 5 0. Since cos u 5 tan u, tan u is undefined when the denominator is zero. c. I c. II 270° 180° (21, 0) (20.7, 0.7) (0.7, 20.7) (20.7, 0.7) b. 6 !5 3 c. 23 Developing Skills Applying Skills 24. a. (5 cos u, 5 sin u) b. (25 cos u, 25 sin u) c. mROP9 5 u, mROP0 5 180 1 u 25. a. (2cos u, sin u) b. u c. 180 2 u 3. a. 45 b. 35 c. 54 d. 34 4. a. 212 13 5 b. 13 c. 212 13 d. 212 5 8 5. a. 10 b. 235 c. 54 d. 243 6. a. 2 !15 4 7. a. 313 1 2 b. 214 b. 2 !3 2 c. 2 !15 4 c. 1 2 d. !15 d. 2 !3 3 14580AK03.pgs 3/27/09 8. a. 32 9. a. 2 !2 2 10. a. 22 !6 5 10:53 AM b. !5 3 b. 2 !2 2 b. 1 5 Page 314 c. 23 c. 2 !2 2 c. 22 !6 5 d. 2 !5 5 d. 1 b. 43 c. !7 4 d. !7 3 13. a. 25 7 b. 25 c. 24 25 d. 24 7 14. a. 17 8 b. 17 c. 15 17 d. 15 8 b. 2 !7 3 c. !2 3 Writing About Mathematics 1 1. Since sec u 5 cos u , for sec u to equal one-half, cos u has to equal 2, which is not possible. 908 0 2. cot 908 5 tan1908 5 cos sin 908 5 1 5 0 d. 22!6 11. a. !7 4 12. a. !2 3 9-5 The Reciprocal Trigonometric Functions (pages 377–378) d. 2 !14 7 Developing Skills 3. a. 0.8 c. 0.75 d. 53 5 1.6 e. 1.25 f. 43 5 1.3 15. a. !2 b. 2 !2 2 c. 2 !2 2 16. a. 5 b. 235 c. 245 d. 43 b. c. d. 22 e. d. 23 5. a. 17. a. 3 !5 18. a. 2 !10 19. a. 4 !2 20. a. 3 !10 2 !5 5 b. 2 !10 10 b. !2 2 b. 3 !10 10 22. IV 25. II 2 !5 5 c. 3 !10 10 c. 2 !2 2 d. 1 4. a. 20.28 c. c. 2 !10 d. 213 10 23. III 26. III 32. m 5 5 sin u cos u d. 25 24 5 1.0416 < 23.5714 f. 224 7 < 23.4286 b. 2 !2 3 d. 3 !2 4 e. 3 f. 2!2 9. a. 3 !2 5 c. 23 !14 7 e. 5 !2 6 10. a. 237 c. 3 !10 20 e. 273 b. 212 f. 2 !3 3 b. 2 !5 3 d. 23 !5 5 f. !5 2 b. 2 !7 5 d. 25 !7 7 f. 2 !14 6 b. 22 !10 7 d. 27 !10 20 f. 2 !10 3 11. a. 5 b. 45 c. 35 d. 34 12. a. !17 c. !17 314 f. 2 !35 35 7. a. 13 c. !2 4 c. 2 !5 5 e. 232 Hands-On Activity 3. Answers will vary: 0.34 5. The values are about the same. 6. Answers will vary: 70° (1, 2.75); 100° (1, 25.67); 165° (1, 20.27); 200° (1, 0.36); 250° (1, 2.75); 300° (1, 21.73); 345° (1, 20.27) In each case, the value of the tangent is approximately equal to the y-coordinate of P. d. 26 d. 22 8. a. 223 5 tan u b. 216 6. a. !3 2 c. 2!3 e. 22 !3 3 Applying Skills 30. a. {u : u is any degree angle} b. {y : 21 # y # 1} c. No, tan u is undefined at 90° 1 180°n. d. {u : u 90° 1 180°n} e. {all real numbers} 31. Apply the Pythagorean Theorem with x 5 cos u and y 5 sin u. b. 0.96 5 20.2916 e. 6 !35 35 21. I 24. I 27. a. 0 b. Undefined c. Answers will vary: 270° 1 360°n 28. a. 0 b. 61 c. Answers will vary: any multiple of 180° 29. a. 61 b. 0 c. Answers will vary: 90° 1 180°n y x 7 224 225 7 !35 6 c. 2!35 d. 21 b. 0.6 b. !17 4 d. 14 14580AK03.pgs 3/27/09 10:53 AM 13. a. 3 !2 Page 315 b. 2!2 c. 2!2 d. 1 14. a. 5 !2 b. 2!2 c. 2!2 d. 1 15. a. 6 !2 b. !2 15. !2 16. !2 2 17. !2 18. 21. 24. 27. 30. 19. 22. 25. 28. 31. 20. 23. 26. 29. 32. 1 21 0 Undefined Undefined c. 2!2 d. 21 33. 1 16. a. 4 !5 b. !5 2 36. 1 c. 2!5 d. 212 17. a. 9 !2 c. !2 b. 2!2 d. 21 18. a. 3 !10 b. b. b. 23. b. 7.5 mi 44. 14 1 12 5 !3 2 1 2 !3 2 ✘ 9-7 Function Values from the Calculator (pages 384–385) c. 12 mi 1 sin u cos u Writing About Mathematics 1. tan 90° is undefined. 2. 400° and 40° are co-terminal angles. Developing Skills 3. 0.4695 6. 20.1736 9. 0.3640 12. 20.9848 15. 0.2679 18. 20.1736 21. 0.9500 24. 0.9621 27. 20.4258 30. 3.8637 33. 1.2208 36. 0.2867 39. 20° 42. 40° 45. 3° u 5 cos sin u , sin u 2 0 y 1. Yes. If P9 5 (x, y), then cos u 5 xr and sin u 5 r . 2. Yes. sin2 u 1 cos2 u 5 1 sin2 u 1 a2 5 1 12. !3 43. 1 1 cos (308) 5 !3 2 . cos (608) 5 2 Writing About Mathematics 9. 2 42. 47. Answers will vary. Example: 9-6 Function Values of Special Angles (pages 380–381) 6. 41. 1 1 2 26. a. 12 mi 1 28. cot u 5 tan u 5 !3 3 40. 2 !3 3 c. 1, 21 c. 1, 21 c. 0 27. a. Divide the given equation by cos2 u. b. No. It is true only where tan u and sec u are defined. c. OT 3. !3 2 39. 1 308 , 608 5 12 Developing Skills 38. 41 1 2 ? sin (308) 1 sin (308) 5 sin (608) 0 0 1, 21 90° 13 b. 12 c. 35. 3 45. 120 ft 3 120 !3 ft 46. Answers will vary: d. 23 19. a. 1, 21 20. a. 1, 21 21. a. 0 22. 1, 21 Applying Skills 24. 7.5 feet 25. a. 13 5 34. !2 21 1 37. 1 21 Undefined 0 21 1 Applying Skills b. 2!10 c. !10 3 1 0 Undefined 21 0 sin u 5 6"1 2 a2 4. 21 7. !3 10. !3 2 13. !3 3 5. 2 8. 21 11. 2 !3 3 14. !2 2 315 4. 7. 10. 13. 16. 19. 22. 25. 28. 31. 34. 37. 40. 43. 46. 0.8192 0.1736 20.2588 0.9848 20.5736 20.1736 0.8450 20.1352 16.1190 0.5095 23.7321 1.6616 64° 35° 62° 5. 8. 11. 14. 17. 20. 23. 26. 29. 32. 35. 38. 41. 44. 47. 4.7046 0.3640 0.2588 0.2679 20.5736 20.1736 38.1885 20.9048 3.2361 25.7588 21.1034 24.4454 12° 87° 33° 14580AK03.pgs 48. 51. 54. 57. 3/27/09 57° 15° 309 14° 159 82° 159 10:53 AM 49. 52. 55. 58. Page 316 46° 74° 309 40° 489 5° 069 50. 85° 53. 75° 459 56. 49° 129 5. a–c. 320° Applying Skills 59. 17° 60. 20° 299 61. 51° 099, 47° 609, 80° 609 62. a. II 63. a. IV b. 143° b. 286° d. 40° y 40° 9-8 Reference Angles and the Calculator (page 391) 6. a–c. d. 45° y Writing About Mathematics 1. Yes, 2u is equivalent to 360 2 u. 2. No. Only sin and tan are negative in quadrant IV. Cos will return a positive value. x 45° 245° Developing Skills 3. a–c. 45° d. 60° y 120° 60° 60° x 40° 7. a–c. x d. 45° y 45° x 405° 4. a–c. d. 70° y 250° 70° 8. 11. 14. 17. 20. 23. 26. 29. 32. 35. 38. 41. 70° x 316 80° 70° 85° 35° tan 75° 2sin 75° 2sin 56° 51°, 309° 54°, 126° 23°, 337° 138°, 222° 0°, 180° 9. 12. 15. 18. 21. 24. 27. 30. 33. 36. 39. 42. 5° 75° 70° 2sin 35° cos 48° 2cos 65° sin 40° 18°, 198° 183°, 357° 96°, 276° 188°, 352° 90°, 270° 10. 13. 16. 19. 22. 25. 28. 31. 34. 37. 40. 43. 30° 50° 50° 2cos 85° 2tan 10° 2tan 55° 20°, 160° 55°, 235° 108°, 252° 14°, 166° 172°, 352° 0°, 180° 14580AK03.pgs 3/27/09 10:53 AM Page 317 Review Exercises (pages 394–396) 1. a–c. 5. a–c. y 600° 220° 40° 7. a. IV 300° 60° x b. !7 4 c. 0.8 c. 34 e. 253 5 21.6 f. 1.25 g. 243 5 21.3 9. a. II d. 35° y b. 1 5 c. 22 !6 5 145° 35° d. 2 !6 12 35° e. 5 x !6 f. 2512 g. 22 !6 11. a. I b. 4. a–c. d. 80° y c. d. e. 80° 80° f. x g. 2100° 317 8. a. I b. 20.6 d. 20.75 60° 3. a–c. x 6. a. 145° b. 58° d. 60° y 60° 60° x 40° 2. a–c. d. 60° y d. 40° 3 5 4 5 3 4 5 3 5 4 4 3 d. !7 3 e. 4 !7 7 f. 43 g. 3 !7 7 10. a. III b. 2 !5 3 c. 223 d. !5 2 e. 23 !5 5 f. 232 g. 2 !5 5 12. a. Quadrantal b. 0 c. 21 d. 0 e. Undefined f. 21 g. Undefined 14580AK03.pgs 3/27/09 10:53 AM 13. a. III b. Page 318 Cumulative Review (pages 396–398) 14. a. IV 235 !10 21350 b. c. 245 c. 9 !10 50 d. 43 d. 213 9 Part I 1. 1 4. 4 7. 1 10. 4 Part II 11. (3 2 2i)(21 e. 25 !10 13 e. 235 f. 245 f. 5 !10 9 g. 43 9 g. 213 15. a. (0.7, 0.7) 12. 23 1 b. (0.2, 1.0) 16. !3 2 17. !3 2 18. 1 23. !3 2 24. !2 20. 2 !2 2 19. 2!3 22. !3 2 21. 2 !3 2 26. !3 3 25. 2 1 2 27. 22 0.6428 29. 5.6713 30. 0.9397 1.1918 32. 20.9925 33. 20.4067 20.6428 35. 0.1763 36. 2cos 80° 2sin 60° 38. cos 80° 39. tan 30° sin 30° 41. 2cos 75° 42. cos 40° tan 50° 44. 22°, 158° 45. 24°, 336° 54°, 234° 47. 224°, 316° 48. 150°, 330° 145°, 215° 50. 44°, 316° 51. 19°, 161° 90°, 270° 53. 0°, 180° 54. 90°, 270° 0°, 180° a. q b. p c. t q t d. By similar triangles, p 5 1 . In parts a–c, we showed that q 5 sin u, p 5 cos u, and sin u t 5 tan u. It follows that tan u 5 cos u by substitution. 57. 6° 509 58. 18 feet the equation of the circle is: (x 2 2)2 1 (y 2 1)2 5 5 Part IV 15. Answer: x 5 3 6 !5 Write the equation in standard form and then use the quadratic formula with a 5 1, b 5 26, and c 5 4: x5 In OTR: x5 OT x5 OT sec u 5 cos u 5 adj 5 OR 5 1 5 OT OS OS adj QS QS 2(26) 6 !(26) 2 2 4(1)(4) 2(1) 6 6 !36 2 16 2 6 6 !20 5 6 6 22 !5 5 3 6 2 !5 16. a. f + g(22) 5 f(5(22) 1 7) 5 f(23) 5 (23) 2 1 3 5 12 b. f + g(x) 5 f(5x 1 7) 5 (5x 1 7) 2 2 5x 2 7 5 25x2 1 70x 1 49 2 5x 2 7 5 25x2 1 65x 1 42 Since QS || OR, mu 5 mQSO. Thus: hyp x x23 x2 2 6x 1 9 x2 2 7x 1 6 1, 6 Check x 5 6: 1 ? 6 3 1 (6 1 3) 2 5 1 ? 6 3 1 (9) 2 5 6 5 6✔ r 5 "(4 2 2) 2 1 (1 2 0) 2 5 "22 1 12 5 !5, Exploration (page 396) hyp 3. 2 6. 4 9. 1 1 i) 5 23 1 3i 1 2i 1 2 5 21 1 5i 2x 2 4 , 3 , 2x 2 4 , 3 , 2x ,7 , x , 72 Part III 13. Answer: x 5 6 1 3 1 (x 1 3) 2 5 1 (x 1 3) 2 5 x135 05 x5 Check x 5 1: 1 ? 3 1 (1 1 3) 2 5 1 1 ? 3 1 (4) 2 5 1 5 21✘ 14. Since 28. 31. 34. 37. 40. 43. 46. 49. 52. 55. 56. 1 2. 3 5. 2 8. 3 csc u 5 csc /QSO 5 opp 5 OQ 5 1 5 OS cot u 5 cot /QSO 5 opp 5 QO 5 1 5 QS 318 14580AK04.pgs 3/26/09 12:06 PM Page 319 Chapter 10. More Trigonometric Functions 10-1 Radian Measure (pages 404–406) Writing About Mathematics 1. Yes, when the length of the intercepted arc is divided by the radius of the circle, the units cancel, giving equivalent ratios. 2. 4p. Two full revolutions is 720° 5 4p radians. Developing Skills 3. p6 4. p2 5. p4 6. 2p 3 7. 8p 9 8. 3p 4 9. 5p 4 10. 4p 3 11. 3p 2 12. 11p 6 13. 60° 14. 20° 15. 18. 21. 23. 16. 19. 22. b. 17. 200° 20. 330° 18° 270° 630° a. 60° c. 72° 540° 57.3° b. p9 25. a. 200° c. y 200° p 9 x b. 7p 18 26. a. 270° c. y p 3 y x 270° p 60° ( 3 ) 7p 18 x b. 2p 9 27. a. 500° c. 24. a. 35° c. y b. 7p 36 140° y 40° x 35° ( 36 ) 7p x 28. 31. 34. 37. 40. 319 6 3.2 5 6p 1.5 m 29. 32. 35. 38. 2 6 4p 3.4 in. 30. 33. 36. 39. 25 40 15 p 2.4 14580AK04.pgs 41. 3/26/09 30° 12:06 PM 45° 60° p 6 p 4 90° 180° p 2 p 3 p Page 320 270° 360° 3p 2 2. 6. 9. 12. 15. 18. 21. 24. 1 1 1.3086 0.5976 1.3785 1.3750 7. 10. 13. 16. 19. 22. 25. 212 2 0.2771 0.3514 0.9912 1.2542 1 2 !3 1 sin 5p 6 B 5 Q2 2 , 2 R ; A cos 7p 6, !3 1 sin 7p 6 B 5 Q2 2 , 22 R ; A cos 2p 3, !3 1 sin 2p 3 B 5 Q22 , 2 2 R ; A cos 5p 4, A cos 3p 2, A cos 5p 3, Writing About Mathematics 1. Yes, p 5 180° so the formula is correct. 2. Yes. Adding any multiple of 2p 5 360° keeps the terminal side the same. Developing Skills 4. !3 A cos 5p 6, !2 !2 sin 3p 4 B 5 Q2 2 , 2 R ; ( cos p, sin p) 5 (21, 0) ; 10-2 Trigonometric Function Values and Radian Measure (pages 409–410) 3. 1 !3 sin 2p 3 B 5 Q22 , 2 R ; A cos 3p 4, 2p Applying Skills 42. 7.2 ft 43. 9 44. a. Yes, one complete revolution for any circle is 2p radians. b. No. The radian measure is the same but the length of the radius is not, so the measure of the arc, and therefore the distance traveled, will differ accordingly. 45. 2,457.4 km !2 2 !3 3 A cos 2p 3, A cos 7p 4, !2 !2 sin 5p 4 B 5 Q2 2 , 2 2 R ; sin 3p 2 B 5 (0, 21) ; !3 1 sin 5p 3 B 5 Q 2, 2 2 R ; !2 !2 sin 7p 4 B 5 Q 2 , 2 2 R; A cos 11p 6 , !3 1 sin 11p 6 B 5 Q 2 , 22 R Hands-On Activity 2: Evaluating the Sine and Cosine Functions 5. 0 9 11 x 1. sin x: x9! , 211! 8. 2 !3 2 11. Undefined 14. 1.0029 17. 0.9732 20. 0.5796 23. 0.4404 26. 0 8 10 x cos x: x8! , 210! 2. 0.7071 3. 0.5000 10-3 Pythagorean Identities (pages 413–414) 27. 1 2 !2 28. 25 !3 2 6 Applying Skills 29. (20.4236, 20.9059) 30. a. 2.50 b. 2.50 c. (20.8011, 0.5985) d. (22.4034, 1.7954) e. Since for both points, x is negative and y is positive, P and B are both in quadrant II. 31. a. 16623 b. 20.1615 32. 28.0 ft 33. a. 2,405 ft b. 2,352 ft Hands-On Activity 1: The Unit Circle and Radian Measure 1. (cos 0, sin 0) 5 (1, 0); Writing About Mathematics 1 1 1 1. Yes. ( sec u)(csc u) 5 cos u ? sin u 5 cos u sin u 2. Yes. Both equations are equivalent to the identity cos 2 u 1 sin 2 u 5 1. Developing Skills !6 3. sin u 5 15 , cos u 5 22 !6 5 , tan u 5 2 12 , !6 cot u 5 22!6, sec u 5 2512 , csc u 5 5 !7 3 !7 3 4. sin u 5 !7 4 , cos u 5 4 , tan u 5 3 , cot u 5 7 , sec u 5 43 , csc u 5 4 !7 7 !7 3 5. sin u 5 2 !7 4 , cos u 5 24 , tan u 5 3 , 4 !7 4 cot u 5 3 !7 7 , sec u 5 23 , csc u 5 2 7 A cos p6 , 1 sin p6 B 5 A !3 2 , 2B; 2 !5 6. sin u 5 223 , cos u 5 !5 3 , tan u 5 2 5 , A cos p3 , sin p3 B 5 A 12, !3 2 B; 2 !5 7. sin u 5 23 , cos u 5 2 !5 3 , tan u 5 2 5 , A cos p4 , A cos p2 , !2 sin p4 B 5 A !2 2 , 2 B; 3 !5 3 cot u 5 2 !5 2 , sec u 5 5 , csc u 5 22 sin p2 B 5 (0, 1) 3 !5 3 cot u 5 2 !5 2 , sec u 5 2 5 , csc u 5 2 320 14580AK04.pgs 3/26/09 12:06 PM Page 321 !5 8. sin u 5 2 !5 5 , cos u 5 2 5 , tan u 5 22, 10-5 Inverse Trigonometric Functions (pages 423–425) cot u 5 212 , sec u 5 2!5, csc u 5 !5 2 9. Writing About Mathematics 1. No. The restricted domain of cosine is 0 # x # p, while the restricted domain of tangent is 2p2 , x , p2 . These two intervals are not equivalent. 2. Yes. The calculator returns an equivalent answer for cos21 (20.5) regardless of whether it is in degrees or radians. Developing Skills 3. a. 30° 4. a. 45° 5. a. 0° b. p6 b. p4 b. 0 6. a. 245° 7. a. 120° 8. a. 230° b. 2p4 b. 2p b. 2p6 3 9. a. 260° 10. a. 260° 11. a. 180° b. 2p3 b. 2p3 b. p 12. a. 90° 13. a. 0° 14. a. 90° b. p2 b. 0 b. p2 15. 37° 16. 127° 17. 77° 18. 277° 19. 26° 20. 154° 21. 46° 22. 246° 23. 287° !17 sin u 5 24 !17 17 , cos u 5 2 17 , tan u 5 4, cot u 5 14 , sec u 5 2!17, csc u 5 2 !17 4 1 10. sin u 5 3 !7 8 , cos u 5 28 , tan u 5 23 !7, 8 !7 cot u 5 2 !7 21 , sec u 5 28, csc u 5 21 5 !34 3 11. sin u 5 23 !34 34 , cos u 5 2 34 , tan u 5 5 , !34 cot u 5 53 , sec u 5 2 !34 5 , csc u 5 2 3 12. sin u 5 45 , cos u 5 235 , tan u 5 243 , cot u 5 234 , sec u 5 253 , csc u 5 54 13. sin u 5 45 , cos u 5 235 , tan u 5 243 , cot u 5 34 , sec u 5 253 , csc u 5 54 6 !37 1 14. sin u 5 2 !37 37 , cos u 5 37 , tan u 5 26 , cot u 5 26, sec u 5 !37 6 , csc u 5 2!37 15. 1 18. 1 sin u 21. 1 16. cos u 19. 1 1 sin u cos u 24. !2 2 17. sin u 20. sin u cos u 22. 0 28. 30. 21 31. p 4 34. 36. p 37. 2!3 3 !2 2 p 6 2p4 x 39. a. u 5 arccos A x 1 1B 3 b. u 5 arcsin A 2x 1 3 B Writing About Mathematics 26. 0 29. 21 32. !2 2 35. p4 38. p6 1 c. u 5 arctan A xx 1 1 2B Applying Skills u 1. Yes, cot u 5 cos sin u 5 cos u ? csc u. Both functions are undefined for integer multiples of p. cos p 2 sin p 2 27. 21 33. 10-4 Domain and Range of Trigonometric Functions (page 419) 2. No, cot p2 5 25. 1 40. a. S 0, p2 B < A p2 , p T b. No. The restricted domain of secant does not include p2 . c. (2`, 21] [1, `) d. (2`, 21] [1, `) e. S 0, p2 B < A p2 , p T 5 01 5 0. Developing Skills 3. 1 4. 0 5. Tangent is undefined at p2 1 np (n 5 0). 6. Secant is undefined at p2 1 np (n 5 0). 7. 1 8. 0 9. 0 10. Cotangent is undefined at np (n 5 1). 11. Secant is undefined at p2 1 np (n 5 1). 12. 21 13. 0 14. Cotangent is undefined at np (n 5 0). 15. Tangent is undefined at p2 1 np (n 5 21). 16. 21 17. Secant is undefined at p2 1 np (n 5 5). 18. Cotangent is undefined at np (n 5 28). 19. Answers will vary: p2 1 np 20. Answers will vary: np 41. a. S 2p2 , 0 B < A 0, p2 T b. No. The restricted domain of cosecant does not include 0. c. (2`, 21] [1, `) d. (2`, 1] [1, `) e. S 2p2 , 0) < A 0, p2 T < A 0, p2 B b. No. The restricted domain of the tangent function does not include 0. 42. a. 321 A 2p2 , 0 B 14580AK04.pgs 3/26/09 12:06 PM Page 322 30. a. sec A 2p3 B c. (2`, `) d. (2`, `) e. A 2p2 , 0 B < A 0, p2 B 43. a. u 5 arctan d1 b. 2 32. a. b. b. 63° 34. a. Writing About Mathematics 1. Yes. Cofunctions allow you to express any trigonometric function in terms of the sine function. Also, reference angles allow you to express any trigonometric function value in terms of an acute angle. 2 uB sec u 5 csc A p2 2 u B 26. a. cos p6 b. !3 2 28. a. cot p3 b. !3 3 cot u 5 tan b. 212 1. 5p 12 2. 3p 4 3. 5p 4 4. 2p3 5. 7. 9. 11. 13. 15. 17. 18. 2. No. If A is in quadrant II, cos A 5 Developing Skills 3. a. cos 25° 4. a. sin 10° b. 0.9063 b. 0.1736 5. a. cot 36° 6. a. cos 4° b. 1.3764 b. 0.9976 7. a. sec 42° 8. a. csc 15° b. 1.3456 b. 3.8637 9. a. tan 33° 10. a. sin 20° b. 0.6494 b. 0.3420 11. a. cos (220°) or cos 20° 12. a. cot (25°) b. 0.9397 b. 211.4301 13. a. sin (240°) 14. a. csc (235°) b. 20.6428 b. 21.7434 15. a. cos (2140°) 16. a. sin (2165°) b. 20.7660 b. 20.2588 17. a. cot (2147°) 18. a. sec (2176°) b. 1.5399 b. 1.0024 19. a. sin (2210°) or sin 30° 20. a. cos (2205°) b. 0.5 b. 20.9063 21. a. tan (2222°) b. 20.9004 22. a. csc (2195°) or csc 15° b. 3.8637 23. 35° 24. 20° 25. cos u 5 sin A p2 2 u B sin u 5 cos A p2 2 u B tan u 5 cot 33. a. sin A 213p 6 B Review Exercises (pages 430–431) 2!3 2 . A p2 b. Undefined cos 3p 4 2 !2 2 cot 13p 6 b. !3 10-6 Cofunctions (pages 427–428) A p2 31. a. tan A 2p2 B 45° 210° 6. 8. 10. 12. 14. 16. p 2 s 5 10 cm r 5 4 cm r 5 4 cm s 5 2p ft a. 90°, 270° c. 90°, 270° 72° 222.5° u51 u 5 1.5 u 5 30 r 5 2.5 cm b. 0°, 180°, 360° d. 0°, 180°, 360° 19. a. Domain: [21, 1], Range: S 2p2 , p2 T b. Domain: [21, 1], Range: f0, pg c. Domain: {all real numbers}, Range: A 2p2 , p2 B 20. 12 21. !2 22. !3 2 23. 1 26. 212 29. 2 32. 213 24. 212 25. 21 27. 28. 22 2 !3 2 30. 0 33. 12 35. 2 !3 3 38. 2!2 31. 21 34. 2 !3 2 36. Undefined 39. 12 Exploration (page 431) 1. y S Q P 2 uB R csc u 5 sec A p2 2 u B O 27. a. sin p4 x T b. !2 2 29. a. csc A 2p6 B 2OT 5 sec u: Let T 5 (0, 2t). b. 22 322 37. !2 14580AK04.pgs 3/26/09 12:06 PM Page 323 Cumulative Review (pages 431–433) Let T9 5 the image of T about a reflection in the x-axis. Then OT 5 OT9 and ROT9 5 180 2 u is a firstquadrant angle. From the Chapter 9 Exploration, Part I 1. 3 4. 3 7. 4 10. 3 Part II sec (180 2 u) 5 OTr 5 OT. Using the properties of reference angles, sec u 5 2 sec (180 2 u) 5 2OT. OS 5 csc u: Let S9 5 the image of S about a reflection in the y-axis. Then OS 5 OS9 and ROS9 5 180 2 u is a first-quadrant angle. From the Chapter 9 Exploration, 11. x 5 2. 4 5. 4 8. 4 6 6 !36 2 4(1)(13) 2(1) x 5 6 6 2!216 x 5 3 6 2i p 12. 2808 ? 180 5 14p 9 Part III 13. Answer: (x 2 2)2 1 (y 2 2)2 5 80 csc (180 2 u) 5 OSr 5 OS Using the properties of reference angles, The radius of the circle 5 "(22 2 6) 2 1 (4 2 0) 2 csc u 5 csc (180 2 u) 5 OS. 2QS 5 cot u: Let S9 5 the image of S about a reflection in the y-axis. Then QS 5 QS9 and ROS9 5 180 2 u is a firstquadrant angle. From the Chapter 9 Exploration, 5 "82 1 42 5 4!5 0 The center of the circle 5 A 22 21 6, 4 1 2 B 5 (2, 2) cot (180 2 u) 5 QSr 5 QS Using the properties of reference angles, 14. a. log13 cot u 5 2cot (180 2 u) 5 2QS. A similar procedure can be used to prove steps 2 and 3. b. 1 Part IV 2. S 1 9 1 2 A 27 B 1 1 ? 243 5 log13 A 9 729 ? 243 B 5 log13 3 !3 1 1 15. cos u 5 sec u 5 !3 5 3 y Q 3. 2 6. 3 9. 2 2 sin u 5 2"1 2 cos2 u 5 2#1 2 A !3 5 2 !6 3 3 B T tan u 5 2" sec2 u 2 1 5 2" A !3B 2 2 1 5 2!2 R O x 16. 1 P 21O 3. S y Q y (1, 22), (4, 1) R O x P T 323 x 14580AK04.pgs 3/26/09 12:06 PM Page 324 Chapter 11. Graphs of Trigonometric Functions 11-1 Graph of the Sine Function (pages 440–441) c. Writing About Mathematics 1. Yes, since for each (x, y) on the graph there is also a point (2x, 2y) on the graph. 2. Yes. The period of y 5 sin x is 2p. 25 6 Developing Skills 3. Graph 5p 7p a. 0 # x , p2 , 3p 2 , x , 2 , 2 , x # 4p 5p 7p b. p2 , x , 3p 2, 2 , x , 2 b. 1 2np or 2p 3 11. a. 2p2 # x # p2 b. 2p # x # p c. 1 2np y e. 10. a. Y2(x) 0.5 0.50000213 0.49999999 < 0.70710678 0.70714305 0.70710647 !2 2 0 Y3(x) 0.52404391 20.0752206 11-2 Graph of the Cosine Function (pages 445–447) x R Writing About Mathematics 1. Yes. For every (x, y) on the graph there is also a point (2x, y) on the graph. 2. Yes. The period of y 5 sin x is 2p. Developing Skills 3. Graph a. p , x , 2p, 3p , x , 4p b. 0 , x , p, 2p , x , 3p c. 2 cycles 4. 1 5. 21 6. 2p 7. No. It fails the horizontal line test. P9 d. sin x Y3 always gives the better approximation. P u O 2u c. x p 6 p 1 u d. 25 sin 5p 12 < 24.15 ft p 4 Applying Skills 9. a–b. 5p 12 p 12 c. 2 cycles 4. 1 5. 21 6. 2p 7. No. It fails the horizontal line test. 8. a. cos p3 5 12 p 3 h 2u is the reflection of u about the x-axis. sin u 5 2sin(2u) Yes. For all angles in the four quadrants, sin u 5 2sin(2u). Yes. sin 0° 5 2sin (20°) 5 sin 180° 5 2sin (2180°) 5 0 sin 90° 5 2sin (290°) 5 sin 270° 5 2sin (2270°) 5 1 Yes, since for all x in the domain, f(x) 5 2f(2x). p 5p 12 # u # 12 b. h(u) 5 25 sin u 324 14580AK04.pgs 3/26/09 12:06 PM Applying Skills 8. a–b. Page 325 11-3 Amplitude, Period, and Phase Shift (pages 453–455) y Writing About Mathematics 1. Yes. The first graph is shifted p2 to the left and the second is shifted p2 to the right, resulting in the graphs starting p units apart. Since this is equal to the period of each curve, their graphs will completely overlap. 2. No. If we factor out a 2 from the second equation, we see that its graph is shifted p8 units, in contrast to a shift of p4 units for the first graph. Developing Skills 3. 1 4. 2 5. 5 P 1 x u O 2u R P9 c. d. e. 9. a. b. 2u is the reflection of u about the x-axis. cos u 5 cos (2u) Yes. For all angles in the four quadrants, cos u 5 cos (2u). Yes. cos 0° 5 cos (20°) 5 cos 180° 5 cos (2180°) 5 1 cos 90° 5 cos (290°) 5 cos 270° 5 cos (2270°) 5 0 Yes. For all x in the domain, f(x) 5 f(2x). d(u) 5 6 cos u p p 36 # u # 18 c. 7. 34 8. 21 10. 18 11. 2p 6. 3 9. 0.6 12. 2p 13. 15. 4p 16. 6p 17. 4p 3 18. 8p 3 20. p2 21. 2p3 19. 2p2 22. p4 24. 23p 4 25. 2p 26. p2 27. d 2p 3 14. p 23. p6 y 6 5.98 1 5.96 5.94 O 5.92 x p 2p 21 5.9 u p 36 p 18 28. p d. 6 cos 18 < 5.901 feet p 10. a. 22 # x # p2 1 3p b. 23p 4 #x# 4 c. x 2p6 cos x !3 2 O Y2(x) Y3(x) < 0.86602540 0.86605388 0.86602526 2p4 !2 2 < 0.70710678 0.70742921 0.70710321 2p 21 y 21 0.12390993 21.211353 Y3 always gives the better approximation. 325 x p 2p 14580AK04.pgs 29. 3/26/09 12:06 PM Page 326 33. y y 4 1 O x p 2 p 2p 3 O 21 30. 24 y 34. 1 O x 2p 0.5 35. x 6p y x O p 2 2p 3 p 21 36. y 3 y 1 2 x O 1 O 4p 1 p 3 21 21 2p 20.5 1 32. x 3p O y O y 4p 21 31. x p 3 x p 2p 4p 21 2p 22 37. y 23 1 x O 21 2 p2 326 p 2 3p 2 14580AK04.pgs 3/26/09 38. 12:06 PM Page 327 Hands-On Activity 1–2. y 5 4 3 2 1 O p 21 2 22 y 1 x O p 4 9p 4 5p 4 21 Applying Skills 39. Since sine and cosine are cofunctions, it follows that sin x 5 cos A x 2 p2 B . 40. a. e 0.014 O 0.05 0.1 0.15 0.2 t a. minimum values: A 5 t O 2 4 21 c. The period of “middle C” appears to be onehalf the period of C3. 42. a. f 84 80 76 72 68 64 4. a. y 5 sin A x 1 p2 B b. y 5 cos x 5. a. y 5 2 sin A x 1 p2 B b. y 5 2 cos x 60 t 4 6 8 52 Writing About Mathematics 1. No. The equation that Tyler wrote has period p and phase shift p. Thus, it is equivalent to y 5 5 cos 2x. The maximum of this curve is at np and the minimum is at p2 1 np. 2. Yes. The phase shift of the first graph is equal to the period of both equations. Developing Skills 3. a. y 5 sin x b. y 5 cos A x 2 p2 B b. 2 (22) 2 (26) 2 11-4 Writing the Equation of a Sine or Cosine Graph (pages 457–459) 8 10 12 14 6 x 3. y 5 3 1 2 sin x 4. Maximum 5 5, minimum 5 1 5. The amplitude is equal to one-half the difference between the maximum and the minimum values: 1 A552 52 2 6. (2) Graph (3) y 5 24 1 2 sin x (4) Maximum 5 –2, minimum 5 –6 (5) The amplitude is equal to one-half the difference between the maximum and 20.014 b. 0.014 volt 41. a–b. h 1 3p 2 10 6. a. y 5 3 sin 2x b. y 5 3 cos A 2x 2 p2 B 7. a. y 5 sin x2 b. No. The period of this function is not 12 2p < 12.566 months. Extending months, but 0.5 this model would shift all the temperatures by more than half a month each year. b. y 5 cos 12 (x 2 p) 8. a. y 5 2 sin A 3x 1 p2 B b. y 5 2 cos 3x 327 14580AK04.pgs 3/26/09 12:06 PM Page 328 9. a. y 5 sin A x 1 p3 B b. y 5 cos A x 2 p6 B 10. a. y 5 2 b. y 5 2 5. a. y 2 sin A x 1 p3 B cos A x 2 p6 B 1 x 11. a. y 5 sin 2 A x 2 p4 B b. y 5 2 cos 2x 2 p2 b. y 5 cos 2 A x 1 p 2 21 22 12. a. y 5 3 sin 12 A x 1 3p 4 B b. y 5 3 cos 12 A x 2 p4 B 13. a. y 5 2 sin 2x O b–c. y p 4B 2 14. a. y 5 2 sin A x 2 p2 B b. y 5 2cos x 1 x O 2 p2 Applying Skills 15. a. 0.75 m b. 10 s 1 c. 10 5 0.1 cycle per second 21 p 2 22 d. h(t) 5 0.75 cos A p5 x B e. No; if the amplitude is 0.75, then the maximum height is 0.75 meter. d. They are the same. Applying Skills 6. a. h 5 r tan u b. V 5 13pr3 tan u 7. a. (1) p2 11-5 The Graph of the Tangent Function (pages 462–463) (2) p4 Writing About Mathematics 1. The tangent graph has no maximum or minimum values, the period is p rather than 2p, it has vertical asymptotes, and the range is all real numbers rather than [21, 1]. 2. No; the range is (2`, `). Developing Skills 3. Graph a. p b. Ux: x 2 p2 1 np V c. (2`, `) 4. a–b. y 4 3 2 1 O x p p 3p 21 2 2 22 23 24 (3) AP 5 r tan p4 (4) AB 5 s 5 2r tan p4 (5) Perimeter 5 2(4)r tan p4 b. (1) 2p 5 (2) p5 (3) AP 5 r tan p5 (4) AB 5 s 5 2r tan p5 (5) Perimeter 5 2(5)r tan p5 c. For any regular polygon with n sides circumscribing a circle of radius r, the perimeter is 2nr tan p n. 11-6 Graphs of the Reciprocal Functions (pages 466–467) Writing About Mathematics 1. Cotangent is the reciprocal of tangent. As the value of tan x increases, its reciprocal decreases, so cot x decreases for all values of x for which it is defined. c. 2 328 14580AK04.pgs 3/26/09 12:06 PM Page 329 2. sec x increases from 1 to ` in the interval S 0, p2 B and increases from 2` to 21 in the p p 3p 14. 23p 2 , 22 , 2 , 2 15. 22p, 2p, 0, p, 2p interval A p2 , p T . p p 3p 16. 23p 2 , 22 , 2 , 2 Developing Skills 3. (3) 4. (8) 6. (7) 7. (1) 9. (2) 10. (4) 11. a. y 17. 22p, 2p, 0, p, 2p 18. y 5 cot x and y 5 csc x 19. y 5 tan x and y 5 sec x 20. a. Odd b. Odd c. Even Applying Skills 21. a. a 5 10 sec u b. 2p p p p u 9 9 18 6 5. (6) 8. (5) a 1 O 10.6 c. No x p 2 10.2 11.5 d. Odd 13.1 d. 115.2 ft 11-7 Graphs of Inverse Trigonometric Functions (pages 471–472) Writing About Mathematics 1. Since sin (230°) 5 212 , arcsin A 212 B reference angle of 230° is 30°. 2. No, tan (220°) 5 0.839 1. Developing Skills 3. 30° 4. 60° 5. 6. 60° 7. 290° 8. 9. 260° 10. 135° 11. 12. 245° 13. 0° 14. 15. p2 16. 0 17. p p 3p b. 23p 2 , 22 , 2 , 2 12. a. y O 21 x p 2 18. p3 21. 24. 0 27. 0 30. 0.5 b. 22p, 2p, 0, p, 2p 13. a. 2p 3 19. 2p3 22. p3 p 2 25. 28. 0 31. 2 !2 2 20. Range 5 Uy: 2p2 # y # p2 V y 5 sin x 1 O p 2 Domain 5 Ux: 2p2 # x # p2 V Range 5 5y: 21 # y # 16 x 34. b. y 5 arccos x Domain 5 5x: 21 # x # 16 Range 5 5y: 0 # y # p6 y 5 cos x Domain 5 5x: 0 # x # p6 Range 5 5y: 21 # y # 16 p p 3p b. 23p 4 , 24 , 4 , 4 329 45° 90° 245° 180° p 4 p 3 23. 2p3 26. 0 29. 1 32. 20.5 In 33–35, part a, answers will be graphs. 33. b. y 5 arcsin x Domain 5 5x: 21 # x # 16 y 5 2308. The 14580AK04.pgs 3/26/09 12:06 PM Page 330 35. b. y 5 arctan x Domain 5 5x: x is a real number6 5. Range 5 Uy: 2p2 , y , p2 V y 5 tan x 36. a. u 5 tan 21 x O Domain 5 Ux: 2p2 , x , p2 V Range 5 5y: y is a real number6 Applying Skills y 1 p 3 21 6. d A 100 B 2p 3 y 2 b. u 5 26.6° or u 5 0.46 radians or u 5 26° 349 c. 71° 349 x O 2p 11-8 Sketching Trigonometric Graphs (pages 474–475) Writing About Mathematics 1. Yes. y 5 tan A x 2 2. p 4B 4p 22 is the tangent graph with a phase shift of p4 . Therefore, the asymptotes also have a phase shift of p4 . No. y 5 sin A 2x 2 p4 B 5 sin 2 A x 2 p8 B . The phase shift is p8 , not p4 . 7. y 4 Developing Skills 3. y 2 O p 2p 24 8. y 3 O p 2 x p 22 4. x O x p 2 y 3 O p 23 23 330 4p 3 p 3 7p 3 x 14580AK04.pgs 3/26/09 9. 12:06 PM Page 331 13. y y 2 1 1 x O x 2 p6 O p 3 2p 21 5p 6 22 14. 21 10. p y 1 x O y p 2p 21 4 15. a. y 5 sin A x 1 p3 B b. y 5 cos A x 2 p6 B 3p O p x 2p 16. a. y 5 22 sin A x 1 p3 B b. y 5 22 cos A x 2 p6 B 17. a. y 5 4 sin x 24 b. y 5 4 cos A x 2 p2 B 11. 18. a. y 5 4 sin A x 1 p2 B y b. y 5 4 cos x 19. a. y 5 24 sin A x 2 p3 B b. y 5 4 cos A x 1 p6 B x O 2 p2 p 2 20. a. y 5 22 sin A x 2 p3 B b. y 5 2 cos A x 1 p6 B 21. a. b. 2 y 2 12. y O x O 22 p 331 x p 2 14580AK04.pgs 3/26/09 22. a. 12:06 PM Page 332 b. 2 y f. y 3 2 2 x O p 2 1 3p 2 2p x O 4p 21 22 23. a. 22 23 b. 4 y 2 O 3. a. c. e. x No amplitude {all real numbers} f. p 2 b. p d. Ux: x 2 p2 1 np V 1 p y 22 x 2 p2 p 2 Review Exercises (pages 476–479) c. 3p 2 e. [22, 2] 1. a. 2 b. 2p 3 d. {all real numbers} f. 2 f. y 1 O 21 c. p1 e. [21, 1] 4. a. 1 b. p d. {all real numbers} y x p 3 1 2p 3 x O 4p 3 p 3 22 21 2. a. 3 b. 4p d. {all real numbers} 1 c. 4p e. [23, 3] 5. a. 1 1 c. 2p b. 2p d. {all real numbers} f. e. [21, 1] y 1 x 2p 332 O 21 p 14580AK04.pgs 3/26/09 12:06 PM 6. a. 2 Page 333 1 c. 2p b. 2p d. {all real numbers} f. 24. a. b. 2 y 2 e. [22, 2] y 1 x 2 2p 1 p 21 x O p 22 2p 21 22 25. No. The function y 5 csc x is undefined at x 5 np for integer values of n. 7. a. y 5 sin 2x 26. a. b. y 5 cos 2 A x 2 p4 B P 110 8. a. y 5 22 sin x 85 9. a. y 5 3 sin 12 A x 1 5p 6 B p 1 b. y 5 3 cos 2 A x 2 6 B 60 b. y 5 2 cos A x 1 p2 B 35 10 10. a. y 5 2sin x b. y 5 cos A x 1 p2 B 1 11. (4) 14. (3) 12. (1) 15. p6 13. (2) 16. p2 17. p6 18. 5p 6 19. p4 b. d. 27. a. c. 20. 2p6 1 c. 25 110 mmHg e. 60 mmHg 4.5 ft b. 14 hr y 5 15.5 1 4.5 sin p7 x 21. a. S 2p2 , p2 T b. [21, 1] c. S 2p2 , p2 T Exploration (pages 478–479) d. Graph Answers will vary. 22. a–b. c. 0 y Cumulative Review (pages 479–481) Part I 1. 3 4. 4 7. 2 10. 3 1 x 22p p 3 2 p 2p 21 2. 4 5. 2 8. 1 3. 3 6. 1 9. 1 Part II 11. 2x 2 5 , 7 27 , 2x 2 5 , 7 , 12 22 , 2x 21 , x ,6 p p 3p 23. x 5 23p 2 , x 5 22 , x 5 2 , x 5 2 1 0 333 1 2 3 4 5 6 t 14580AK04.pgs 3/26/09 12:06 PM Page 334 Part IV 12. Answer: 123°, 303° tan u 5 21.54 u 5 tan21 (21.54) u < 2578 Tangent is negative in the second and fourth quadrants. Therefore, u 5 180 2 57 5 1238 and u 5 360 2 57 5 3038. Part III 13. Answer: x 5 0, 4 6 3i x3 2 8x2 1 25x 5 0 x(x2 2 8x 1 25) 5 0 Therefore, x 5 0 is one solution. Use the quadratic formula to find the roots of the quadratic factor: 15. a. Since BG is the diagonal of a square, mGBC 5 45. GC 1 b. tan u 5 AC 5 !2 5 !2 2 21 !2 6 u 5 tan A 2 B < 358 16. Answer: A 12, 26 B and (2, –3) Substitute the linear equation into the quadratic and solve for x: y 5 2x 2 7 2x 2 7 5 2x2 2 3x 2 5 0 5 2x2 2 5x 1 2 0 5 (2x 2 1)(x 2 2) x 5 12, 2 y 5 2 A 12 B 2 7 5 26 x 5 8 6 2!236 x 5 4 6 3i y 5 2(2) 2 7 5 23 5 14. a 3(2) n 2 1 5 3(2) 21 1 3(2) 0 1 3(2) 1 n50 1 3(2) 2 1 3(2) 3 1 3(2) 4 5 94.5 Chapter 12.Trigonometric Identities 12-1 Basic Identities (page 485) 12-2 Proving an Identity (pages 487–488) Writing About Mathematics 1. No. We also need to know the quadrant in which the terminal side of the angle lies to determine the sign of the other trigonometric functions. 2. a. To derive 1 1 tan2 u 5 sec2 u, divide each term of sin2 u 1 cos2 u 5 1 by cos2 u. To derive cot2 u 1 1 5 csc2 u, divide each term of sin2 u 1 cos2 u 5 1 by sin2 u. b. No; tan u and sec u are not defined for u 5 p2 1 np and cot u and csc u are undefined for u 5 np, so the identities are not defined for those values of u. Developing Skills Writing About Mathematics 1. No. The equation is conditional. If u is an angle whose terminal side lies in quadrant III or IV, then the equation is false. 2. Yes. The fraction is equal to 1. When the left side 2u is multiplied and simplified it becomes 1 cos 1 sin u . sin u 3. cos u 1 6. sin u u 4. cos sin u 7. sin1 u 9. sin12 u 12. cos12 u 10. cos1 u cos u 13. 2 sin u Developing Skills ? 3. sin u csc u cos u 5 cos u 1 sin u cos u ? 1 1 ? sin u ? 1 5 cos u 1 cos u 5 cos u ✔ ? 4. tan u sin u cos u 5 sin2 u 5. cos1 u 8. cos12 u 1 sin u sin u cos u ? cos u ? 1 ? 1 5 sin2 u 1 sin2 u 5 sin2 u ✔ 11. sin u cos u u cos u 14. 1 1 sin cos u ? cos2 u 5. cot u sin u cos u 5 1 cos u sin u cos u ? sin u ? 1 ? 1 5 cos2 u 1 cos2 u 5 cos2 u ✔ 334 14580AK04.pgs 3/26/09 12:06 PM Page 335 ? sec u (cos u 2 cot u) 5 1 2 csc u 6. 16. ? 1 2 csc u sec u cos u 2 sec u cot u 5 1 1 u cos u ? 1 ? cos 1 2 cos u ? sin u 5 1 2 csc u 1 cos u 1 1 ? 1 2 sin1 u 5 1 2 csc u 1 2 csc u 5 1 2 csc u ✔ 17. ? 1 1 sec u csc u sin u 1 csc u tan u 5 1 1 sin u ? 1 sin u 1 1 1 sin u 1 ? sin u ? cos u 5 1 1 sec u 1 ? 1 1 cos1 u 5 1 1 sec u 1 1 sec u 5 1 1 sec u ✔ 8. u ? 2 1 2 cos sec u 5 sin u 12 cos u ? 5 1 cos u sin2 u ? 2 12 2 sin u ? 5 1 sin u 2 1 2 cos u 5 sin u sin2 u 5 sin2 u ✔ ? 5 tan u ? 5 tan u ? 5 tan u 18. cos2 u ? sin u 1 sin u cos u 2 cos u sin2 u 1 sin u cos u 2 sin u cos u 1 2 sin2 u sin u cos u cos2 u sin u cos u cos u sin u ? 5 cot u ? 5 cot u ? 5 cot u ? 5 cot u ? 5 cot u ? cot u 5 cot u ✔ sin u ? 2 1 2 csc u 5 cos u 9. ? 5 tan u ? 5 tan u tan u 5 tan u ✔ ? csc u (sin u 1 tan u) 5 1 1 sec u 7. cos u 1 sin u cos u 2 sin u cos2 u 1 sin u cos u 2 sin u cos u 1 2 cos2 u sin u cos u sin2 u sin u cos u sin u cos u 2 1 2 sin u 5 cos u cos2 u 5 cos2 u ✔ ? sin2 u 1 1 cos u 5 1 sin2 u 1 2 cos u ? 1 1 cos u ? 1 2 cos u 5 1 sin2 u (1 2 cos u) ? ? 10. sin u (csc u 2 sin u) 5 cos2 u 2 cos u 5 1 2 cos u 2 1 sin2 2 cos u 1 2 cos u u (1 2 cos u) sin2 u ? 5 1 2 cos u 1 ? cos2 u sin u csc u 2 sin2 u 5 1 2 cos u 5 1 2 cos u ✔ 1 sin u 1 ? ? sin1 u 2 sin2 u 5 cos2 u cos2 u ? 1 1 sin u 5 1 2 sin u cos2 u 1 2 sin u ? 1 1 sin u ? 1 2 sin u 5 1 2 sin u 19. 1 ? 1 2 sin2 u 5 cos2 u cos2 u 5 cos2 u ✔ ? cos2 5 1 2 sin u 2 11. cos u (sec u 2 cos u) 5 sin u 1 cos2 u (1 2 sin u) cos2 u ? sin2 u cos u sec u 2 cos2 u 5 1 cos u 1 u (1 2 sin u) ? 1 2 sin u 2 ? 5 1 2 sin u 1 ? ? cos1 u 2 cos2 u 5 sin2 u 1 2 sin u 5 1 2 sin u ✔ 1 ? tan u 1 cot u 20. sec u csc u 5 ? 1 2 cos2 u 5 sin2 u 2 sin u 5 sin2 u ✔ ? sin u cos u sec u csc u 5 cos u 1 sin u u ? 12. tan sec u 5 sin u u ? 13. cot csc u 5 cos u sin u cos u ? 5 1 cos u cos u sin u ? 5 1 sin u sin u sin u 5 sin u ✔ ? 1 sec u csc u 5 cos u sin u cos u 5 cos u ✔ u ? 15. sec csc u 5 tan u 1 sin u ? 5 1 cos u cot u 1 cos u ? 5 1 sin u tan u cos u ? sin u 5 cot u sin u ? cos u 5 tan u 2 ? sin2 u 1 cos2 u sec u csc u 5 cos u sin u cos u csc u ? 14. sec u 5 cot u cot u 5 cot u ✔ 2 ? sin u cos u sec u csc u 5 cos u sin u 1 sin u cos u sec u csc u 5 sec u csc u ✔ tan2 u sec u 2 1 sec2 u 2 1 sec u 2 1 21. ? 2 15 sec u ? 2 15 sec u 1 (sec u 1 1)(sec u 2 1) sec u 2 1 tan u 5 tan u ✔ ? 2 15 sec u 1 ? sec u sec u 1 1 2 1 5 sec u 5 sec u ✔ 335 14580AK04.pgs 3/26/09 12:06 PM Page 336 ? sin2 u cos u 1 1 1 cos u 5 1 22. 9. 21 12. 2 !2 2 sin2 ? cos u 1 1 cos u u 1 ? 1 1 cos u 1 1 1 cos u 5 1 2 2 ? cos u 1 cos u sin u 1 1 cos u 1 1 1 cos u 5 1 cos u 1 cos2 u 1 sin2 u ? 51 1 1 cos u cos u 1 1 ? 1 1 cos u 5 1 ? 2 cos2 uu 5 1 1 cos2 u sin2 u ? 2 cos 2u 5 1 cos2 u ? cos2 u 5 1 cos u sec u cos u 1 cos u sin u ? 5 1 sin u ? 2 sin2 uu 5 1 cos2 u ? 1 sin2 u 2 sin2 u 5 1 1 2 cos2 u ? 51 sin2 u c. cos (u 2 458) 5 cos u cos 458 1 sin u sin 458 4 !2 5 A 35 B Q !2 2 R 1 A5B Q 2 R !2 5 710 1 1 5 1✔ d. 8° 12-4 Cosine (A 1 B) (pages 495–496) Hands-On Activity 1 3. Draw segment PPr and let R be its point of intersection with the x-axis. Then: sin u 5 RP sin (2u) 5 2RPr 27. It is undefined when sec u or csc u are undefined; that is, at np 2 where n is an integer. In steps 4–6, the procedures will be similar. Writing About Mathematics 1. No. Maggie added the angles, which is incorrect. The correct answer is: Writing About Mathematics 1. Yes, the equations were shown to be true for all real numbers. 2. Yes. She used the identity cos (90 2 B) 5 sin B and she let B 5 100°. Developing Skills 7. 212 cos u 5 OR cos (2u) 5 OR We have shown that cos u 5 cos (2u). Since RP 5 RP9, sin u 5 2sin (2u). 12-3 Cosine (A 2 B) (pages 491–493) 6. 2 !3 2 !2 f. !6 1 4 21. a. 45° b. sin u 5 45 5 0.8 cos u 5 35 5 0.6 cos2 sin2 u ? sin2 u 5 4. 2 !2 2 !2 d. !6 1 4 !2 e. 2 !6 1 4 ? 1 cos2 u 1 sin2 u 5 1 5 1✔ 3. 212 b. 212 !2 c. 2 !6 1 4 sin u ? 1 csc u51 1 !2 f. !6 2 4 20. a. 2 !3 2 1 5 1✔ 26. !6 d. !2 2 4 !2 e. !6 2 4 1 sin u sin u 1 b. !3 2 !2 c. !6 2 4 ? u 2 25. csc sin u 2 cot u 5 1 1 2 sin2 u ? cos2 u 5 !2 d. !6 1 4 19. a. 212 1 5 1✔ 17. 21 !2 b. 2 !6 1 4 !2 c. !6 1 4 sin u 1 1 sin u cos2 u ? 1 ? 1 1 sin u 1 1 1 sin u 5 1 sin u 1 sin2 u cos2 u ? 1 1 sin u 1 1 1 sin u 5 1 sin u 1 sin2 u 1 cos2 u ? 51 1 1 sin u sin u 1 1 ? 1 1 sin u 5 1 sin2 14. 12 !2 18. a. !6 1 4 2 1 cos u cos u 13. 212 Applying Skills u ? sin u 1 1 cos 1 sin u 5 1 ? sec u 2 24. cos u 2 tan u 5 1 11. 2 !3 2 16. 2 !3 2 15. 21 1 5 1✔ 23. 10. !3 2 cos (A 1 B) 1 cos (A 2 B) 5 cos A cos B 2 sin A sin B 1 cos A cos (2B) 2 sin A sin (2B) 5 2 cos A cos B 5. 2 !3 2 8. !3 2 2. Yes. See the answer to Exercise 1. 336 14580AK04.pgs 3/26/09 12:06 PM Page 337 5. sin (180° 1 30°) 5 212 Developing Skills 3. 2 !3 2 4. 2 !2 2 7. 2 !3 2 6. 212 9. 21 10. 12. 212 13. 15. 2 !3 16. 2 Applying Skills 18. a. 19. a. c. 20. a. 5. 212 8. 2 !2 2 !3 2 !3 2 1 2 !6 2 !2 4 212 !6 2 !2 1 4 !2 2 11. 14. 17. b. b. d. b. sin (180° 2 30°) 5 21 !2 2 !6 4 !3 2 !2 1 !6 4 !2 22 6. sin (270° 1 60°) 5 212 !2 2 !2 2 !6 2 !2 4 c. sin (270° 2 60°) 5 212 7. sin (270° 1 30°) 5 2 !3 2 sin (270° 2 30°) 5 2 !3 2 8. sin (60° 1 90°) 5 21 !6 2 !2 4 sin (60° 2 90°) 5 212 9. sin (30° 1 90°) 5 !3 2 sin (30° 2 90°) 5 2 !3 2 !2 c. !6 1 4 10. sin (90° 1 60°) 5 12 d. cos 4058 5 cos (3608 1 458) 5 cos 3608 cos 458 2 sin 3608 sin 458 5 (1)(cos 458) 2 (0)(sin 458) 5 cos 458 sin (90° 2 60°) 5 21 11. sin (60° 1 270°) 5 212 sin (60° 2 270°) 5 12 12. sin (45° 1 270°) 5 2 !2 2 21. a. AB 5 50, sin u 5 35 5 0.6, cos u 5 45 5 0.8 b. cos (u 1 458) 5 cos u cos 458 2 sin u sin 458 5 A 45 B Q !2 2 R 5 !2 10 2 sin (45° 2 270°) 5 !2 2 A 35 B Q !2 2 R 13. sin (30° 1 270°) 5 2 !3 2 sin (30° 2 270°) 5 !3 2 14. sin (360° 1 60°) 5 !3 2 AC c. cos (u 1 458) 5 AD !2 10 d. 280 ft 5 sin (360° 2 60°) 5 2 !3 2 40 AD AD 5 200 !2 < 282.84 ft 15. sin A 3p 2 1 2p B 5 21 3p sin A 2 2 2p B 5 21 p 1 16. sin A 2p 3 1 6B 5 2 p sin A 2p 3 2 6B 5 1 12-5 Sine (A 2 B) and Sine (A 1 B) (pages 498–500) Writing About Mathematics 1. No. William added the angles, which is incorrect. The correct answer is: !6 1 !2 17. sin A p3 1 5p 4 B 5 2 4 !2 2 !6 sin A p3 2 5p 4 B 5 4 Applying Skills sin (A 1 B) 1 sin (A 2 B) 5 sin A cos B 2 cos A sin B 1 sin A cos (2B) 2 cos A sin (2B) 5 2 sin A cos B !2 18. a. !6 2 4 !6 c. !2 2 4 2. Yes. See the answer to Exercise 1. Developing Skills 19. a. !3 2 !2 c. !6 1 4 3. sin (180° 1 60°) 5 2 !3 2 !2 e. 2 !6 1 4 sin (180° 2 60°) 5 !3 2 4. sin (180° 1 45°) 5 2 !2 2 sin (180° 2 45°) 5 20. a. 212 !2 2 !2 c. !6 2 4 !6 e. !2 2 4 337 !2 b. !6 2 4 !6 d. !2 2 4 b. 212 !2 d. !6 1 4 b. 2 !3 2 !6 d. !2 2 4 !2 f. !6 1 4 14580AK04.pgs 3/26/09 12:06 PM Page 338 21. a. sin x 5 17 33 5. tan (60° 1 60°) 5 2!3 tan (60° 2 60°) 5 0 !2 cos x 5 2033 6. tan (180° 1 30°) 5 !3 3 tan (180° 2 30°) 5 2 !3 3 sin y 5 13 cos y 5 2 !2 3 b. sin (x 2 y) 5 sin x cos y 2 cos x sin y 7. tan (180° 1 45°) 5 1 tan (180° 2 45°) 5 21 c. 12° 9. tan (120° 1 30°) 5 2 !3 3 tan (120° 2 30°) 5 undefined 8. tan (180° 1 60°) 5 !3 tan (180° 2 60°) 5 2!3 2 !2 20 !2 1 5 A 17 33 B Q 3 R 2 Q 33 R A 3 B 5 14 99 !2 10. tan (120° 1 45°) 5 !3 2 2 tan (120° 2 45°) 5 2 1 !3 5 22. a. sin u 5 13 , cos u 5 12 13 b. sin (u 1 308) 5 sin u cos 308 1 cos u sin 308 5 12 1 5 A 13 B Q !3 2 R 1 A 13 B A 2 B c. 5 !3 1 12 26 11. tan (120° 1 60°) 5 0 tan (120° 2 60°) 5 !3 5 5 !3261 12 5 x5 12. tan (120° 1 120°) 5 !3 tan (120° 2 120°) 5 0 500 x 13,000 5 !3 1 12 13. tan (240° 1 120°) 5 0 tan (240° 2 120°) 5 2!3 14. tan (360° 1 60°) 5 !3 tan (360° 2 60°) 5 2!3 x < 629.23 ft 23. Answer: ArA2!2, 7 !2B 15. tan A p 1 p3 B 5 !3 tan A p 2 p3 B 5 2!3 r 5 "62 1 82 5 10, cos a 5 35 , sin a 5 45 Therefore, A9 5 (10 cos (a 1 45), 10 sin (a 1 45)) . 5p 16. tan A 5p 6 1 6 B 5 2!3 10 cos (a 1 458) 5 10(cos a cos 458 2 sin a sin 458) 5p tan A 5p 6 2 6 B 5 0 !2 4 5 10 S A 35 B A !2 2 B 2 A5B A 2 B T 17. tan A p3 1 p4 B 5 22 2 !3 tan A p3 2 p4 B 5 2 2 !3 5 10Q2 !2 10 R 5 2!2 10 sin (a 1 458) 5 10(sin a cos 458 1 cos a sin 458) !2 3 5 10 S A 45 B A !2 2 B 1 A5B A 2 B T 5 !2 10Q 710 R Applying Skills 1808 1 tan u 18. tan (180 1 u) 5 1tan 2 tan 1808 tan u tan u 5 1 20 1 (0)( tan u) 5 7!2 5 tan u 12-6 Tangent (A 2 B) and Tangent (A 1 B) (pages 502–504) 19. 1 20. 274 21. cos A 5 2"1 2 (0.6) 2 5 20.8 Writing About Mathematics 1. If A or B is equal to p2 1 np for any integer n, then tan A or tan B is undefined. 2. When A 5 p6 and B 5 p3 , then tan A tan B 5 1. That makes the denominator of the fraction zero and the fraction undefined. Developing Skills sin A tan A 5 cos A 5 20.75 A 1 tan B tan (A 1 B) 5 1tan 2 tan A tan B 1 4 tan (A 1 B) 5 1 20.75 (0.75)(4) tan (A 1 B) 5 22.375 3. tan (45° 1 30°) 5 2 1 !3 tan (45° 2 30°) 5 2 2 !3 A 2 tan B 22. tan (A 2 B) 5 1tan 1 tan A tan B 2 2 (22) tan (A 2 B) 5 1 1 (2)(22) tan (A 2 B) 5 243 4. tan (45° 1 60°) 5 22 2 !3 tan (45° 2 60°) 5 22 1 !3 338 14580AK05.pgs 3/26/09 12:07 PM Page 339 A 1 tan B 23. tan (A 1 B) 5 1tan 2 tan A tan B tan (A 1 B) 5 b. 2119 169 15. a. 120 169 c. 2120 119 232 1 32 1 2 A 232 B A 23 B d. Quadrant II 16. a. 2 !14 9 tan (A 1 B) 5 0 24. a. 1 b. Yes, tan (x 1 y) 5 1 and tan z 5 11 5 1. 4 25. a. 19 b. 12° 10 26. a. tan x 5 45 , tan y 5 52 b. 33 b. 59 c. 2 !14 5 d. Quadrant I b. 35 17. a. 245 c. 243 d. Quadrant IV 5 b. 213 18. a. 212 13 12-7 Functions of 2A (pages 507–508) c. 12 5 Writing About Mathematics 1. No. Let 2u 5 A. Using the cofunction identity, cos A 5 sin (90° 2 A). Then by substitution, cos 2u 5 sin (90° 2 2u). 2. Yes. Let 2u 5 A. Using the quotient identity, sin A tan A 5 cos A . Then by substitution, sin 2u tan 2u 5 cos 2u . 19. a. 235 5. a. 2 !3 2 6. a. 1 b. 0 b. 12 b. 0 7. a. !3 2 b. 12 8. a. b. 212 9. a. c. 10. a. c. 2 !3 2 15 17 15 8 25 !11 18 5 !11 2 7 !6 11. a. 2049 c. 20 !6 12. a. !3 2 c. !3 !10 13. a. 1249 !10 c. 1231 14. a. 245 c. 243 d. Quadrant III b. 235 20. a. 45 c. 243 21. cot u b. 12 b. 245 c. 43 Developing Skills 3. a. !3 2 4. a. 1 d. Quadrant III c. !3 cot u c. 2 !3 cot u cot u c. Undefined d. Quadrant II 2u 5 1 2sincos 2u ? 2 sin u cos u 5 1 2 (1 2 2 sin2 u) ? 2 sin u cos u 5 2 sin2 u ? cos u 5 sin u ? cot u 5 cot u ✔ c. Undefined c. !3 cos 2u sin 2u ? sin u 1 cos u 5 cos u (cos 2u) 1 sin u (sin 2u) ? 5 sin u cos u 22. c. !3 1 8 b. 17 csc u 1 cos u (cos2 u 2 sin2 u) 1 sin u(2 sin u cos u) ? 5 sin u cos u csc u 1 d. Quadrant I b. csc u cos2 u 2 sin2 u 1 2 sin2 u ? 5 csc u sin u cos2 u 1 sin2 u ? 5 csc u sin u 1 ? sin u 5 csc u 7 18 d. Quadrant IV csc u 5 csc u ✔ 1 b. 49 d. Quadrant I 23. cos 2u cos 2u cos 2u cos 2u cos 2u b. 31 49 24. d. Quadrant I b. 12 5 5 5 5 5 2 2 cos u 2 sin u cos2 u 1 sin2 u 2 sin2 u 2 sin2 u (cos2 u 1 sin2 u) 2 (sin2 u 1 sin2 u) 1 2 2 sin2 u cos 2u ✔ ? 1 csc 2u 5 2 sec u csc u ? 1 1 sin 2u 5 2 sec u csc u ? 1 1 2 sin u cos u 5 2 sec u csc u ? 1 1 1 1 2 A sin u B A cos u ) 5 2 sec u csc u 1 1 2 sec u csc u 5 2 sec u csc u d. Quadrant I b. 35 d. Quadrant IV 339 ✔ 14580AK05.pgs 3/26/09 12:07 PM Page 340 12. a. !7 4 25. a. Let 4A 5 2u, then 2A 5 u. sin 2u 5 2 sin u cos u sin 4A 5 2 sin 2A cos 2A b. Let 4A 5 2u, then 2A 5 u. cos 2u 5 cos2 u 2 sin2 u cos 4A 5 cos2 2A 2 sin2 2A c. Let 4A 5 2u, then 2A 5 u. u tan 2u 5 1 22tan tan2 u tan 4A 5 26. a. 13. a. 2 !15 5 14. a. 213 5 cos A 1 2 cos A 5 6#11 2 1 cos A 3 1 2 cos A 5 6# 1 2 cos2 A (1 2 cos A) 2 !15 4 A) 2 5 6#(1 2sincos 2A A 5 61 2sincos A 12-8 Functions of 12 A (pages 511–513) 20. tan 18 5 Writing About Mathematics 1. Yes. Cosine is positive in the first and fourth quadrants, that is, cos A . 0 when 2p2 , A , p2 . 1 2 A is in the first or fourth quadrant p 1 2p4 , A 2 , 4 , so cos 2 A is positive. tan u 5 tan 12A 5 sin 21 A cos 21 A 2 2 !2 2 2 !2 since 4. a. !3 2 5. a. 12 6. a. 0 7. a. 2 !2 2 8. a. 9. a. 10. a. 11. a. !2 2 !2 4 !102 12 !5 3 5 21. tan 158 5 by substitution, (2 2 !3) 2 1 5 $ b. 2 !3 2 b. 21 b. !2 2 b. b. 1 2 !3 2 ä 1 1 !3 2 2 2 !3 2 2 !3 b. 12 b. 21 5 $ 2 1 !3 ? 2 2 !3 . b. 212 b. (2 2 !2) 2 2 2 2 !2 5 !2 !2 5 $ Developing Skills 3. a. 2 !3 2 1 2 !2 2 ä 1 1 !2 2 5 $ 2 1 !2 ? 2 2 !2 5 u. Using the quotient identity, sin u cos u . Then c. 234 cos A 19. tan 12A 5 6#11 2 1 cos A !6 cos u 5 !10 4 , sin u 5 4 , r 5 4 2 6 cos 2u 5 10 16 5 14 2. Yes. Let c. 212 c. 21 1 3!10 Applying Skills 27. x 5 r cos u, y 5 r sin u 1 2A c. 43 5 !10 b. 2"50 2 10 5 c. 180 7 mi or 257 mi As A1, !15B c. !2 4 b. 22 !5 5 b. 45 5 !10 18. a. "50 1 10 7 b. mBAC 5 2u 5 18 sin 2u 5 c. !6 2 b. 45 16. a. !5 5 17. a. 235 5 6 !10 2Q !6 4 RQ 4 R b. 2 !10 5 b. 22 !2 3 15. a. 35 2 tan 2A 1 2 tan2 2A c. 2 !7 3 b. 234 2 !2 2 !14 4 !42 12 223 c. !3 22. sin 158 5 c. !3 5 2 2 !3 1 2 !3 2 É 2 !3 5 $ 2 24 !3 5 "2 2 2 23. a. Let 14A 5 12u, then 12A 5 u. c. 2 !3 3 c. 0 sin 12u 5 6#1 2 2cos u c. 21 1 2 cos 21 A 2 1 1 Let 4A 5 2u, then 12A 5 cos 12u 5 6#1 1 2cos u sin 14A 5 6 c. 21 c. !7 7 b. c. !119 7 c. 2 !5 2 É cos 14A 5 6 340 É 1 1 cos 21 A 2 u. 14580AK05.pgs 3/26/09 12:07 PM Page 341 c. Let 14A 5 12u, then 12A 5 u. 4. cos u tan 12u 5 6#11 2 1 cos u tan 14A 5 6 1 2 cos 21 A 1 ä 1 1 cos 2 A 5 24. a. cos u 5 25 65 5 13 tan 12u 5 5 b. tan 12u 5 2 3 5 h5 5 1 2 13 5 ä 1 1 13 5. 4 2 #9 5 3 height of base 25 h 25 50 2 3 or 163 ft sin A cos B 1 cos A sin B 1 sin A cosB 2 cos A sin B sin A cos B 1 cos A sin B 2 sin A cos B 1 cos A sin B ? 5 tan A cot B 2 sin A cos B ? 2 cos A sin B 5 tan A cot B sin A cos B ? cos A ? sin B 5 tan A cot B tan A cot B 5 tan A cot B ✔ 7. 224 25 1. sec u 5 csc u tan u ? sin u 1 sec u 5 sin u ? cos u 44 12. 2125 13. 53 14. 117 125 15. 34 44 16. 2117 17. 336 625 7 18. 25 25. !7 3 28. 3!7 ? csc u 2. cos u cot u 1 sin u 5 ? cos u cos u sin u 1 sin u 5 csc u 31. !7 7 u ? 1 sin u A sin sin u ) 5 csc u !2 34. 710 7 37. 25 u ? 5 csc u 1 ? sin u 5 35 !2 40. 2"50 2 10 csc u csc u 5 csc u ✔ 2 11. 45 22. 8 !6 5 sec u 5 sec u ✔ 7 9. 24 8. 245 10. 43 19. 336 527 ? 1 sec u 5 cos u u 1 sin u tan u sin (A 1 B) 1 sin (A 2 B) ? sin2 tan u ? tan A cot B 6. sin (A 1 B) 2 sin (A 2 B) 5 Review Exercises (page 514) cos2 ? sin 2u 1 sin u cos 2u 1 cos u 1 1 5 ? 2 sin u cos u 1 sin u 2 cos2 u 2 1 1 cos u 1 1 5 ✔ tan u 5 tan u ✔ Hands-On Activity: Graphical Support for the Trigonometric Identities 1. Yes 2. Yes 3. Yes 4. Each graph, Y2 and Y3, coincides with Y1 only part of the time. When cos x2 is positive, Y2 coincides and when cos x2 is negative, Y3 coincides. Neither Y2 nor Y3 is accurate for all values of x. u sin u 1 cos u cot u 1 cos u cot u sin u cos u ? 1 1 cos u sin u 1 5 cos u cos u cot u sin u(1 1 cos u) ? 1 1 cos u 5 cot u cos u ? 1 1 cos u tan u(1 1 cos u) 5 cot u 1 1 cos u 1 1 cos u 5 cot u cot u sin u(2 cos u 1 1) ? cos u(2 cos u 1 1) 5 tan u sin u ? cos u 5 tan u 1 height of bill board 5 60 2 50 3 5 433 ft cos2 ? 1 tan u 1 csc1 u 5 ? 1 sin u cos u 1 sin u 5 !2 20. 710 21. 2 !2 10 23. 20.8 24. 34 26. 3 !7 8 29. !2 4 9 32. 216 35. !2 10 38. 24 25 27. 18 30. !14 4 33. 5 !2 7 36. 7 7 39. 24 35 !2 41. 2"50 1 10 42. If A and B complementary, cos (A 1 B) 5 cos 90°. cos (A 1 B) 5 cos A cos B 2 sin A sin B cos 908 5 cos A cos B 2 sin A sin B 0 5 cos A cos B 2 sin A sin B sin A sin B 5 cos A cos B ✔ ? 3. 2 sin u 5 1 2 cos 2u ? 2 sin2 u 5 1 2 (1 2 2 sin2 u) 2 2 sin u 5 2 sin2 u ✔ 341 14580AK05.pgs 3/26/09 12:07 PM Page 342 Exploration (page 515) Part II 1 2 i 1 2 i 1 1 1 1 11. 1 1 i 5 1 1 i ? 1 2 i 5 2 5 2 2 2i 12. a 5 52 and c 5 25 4 If x2 1 5x 1 c 5 (x 1 a)2, then x2 1 5x 1 c 5 x2 1 2ax 1 a2. Therefore, 2a 5 5, so a 5 52 and a2 5 c 5 25 4. 1. The equations appear to be identities since when each left side is graphed in Y1 and each right side is graphed in Y2, the graphs of Y1 and Y2 coincide. 2. sin (2A 1 A) 5 sin 2A cos A 1 cos 2A sin A sin (2A 1 A) 5 (2 sin A cos A) cos A 1 (2 cos2 A 2 1) sin A sin (2A 1 A) 5 2 sin A cos2 A 1 2 sin A cos2 A 2 sin A sin (2A 1 A) 5 sin A (4 cos2 A 2 1) sin (2A 1 A) 5 sin A (4 (1 2 sin2 A) 2 1) sin (2A 1 A) 5 sin A (3 2 4 sin2 A) sin (3A) 5 3 sin A 2 4 sin3 A ✔ Part III 13. x2 1 3x 2 10 $ 10 (x 1 5)(x 2 2) $ 0 x # 25 or x $ 2 7 6 5 4 3 2 1 0 cos (2A 1 A) 5 cos 2A cos A 2 sin 2A sin A cos (2A 1 A) 5 (2 cos2 A 2 1) cos A 2 (2 sin A cos A) sin A cos (2A 1 A) 5 (2 cos2 A 2 1) cos A 2 (2 sin2 A) cos A cos (2A 1 A) 5 cos A [(2 cos2 A 2 1) 2 2 sin2 A] cos (2A 1 A) 5 cos A [2 cos2 A 2 1 2 2(1 2 cos2 A)] cos (2A 1 A) 5 cos A (4 cos2 A 2 3) cos (3A) 5 4 cos3 A 2 3 cos A ✔ tan (2A 1 A) 5 tan (2A 1 A) 5 tan (2A 1 A) 5 tan (2A 1 A) 5 tan (3A) 5 2A A 1 2 1 22tan tan2 A B radius: " (2 2 1) 2 1 (5 2 2) 2 5 !10 equation: (x 2 1) 2 1 (y 2 2) 2 5 10 Part IV 15. a. 1 p 2 x b. g(x) 5 3 1 sin A x 1 p4 B 16. a. a5 5 a1r5 2 1 9 5 r4 r 5 !3 b. 1, !3, 3, 3!3, 9, 9!3, 27, 27!3 2 3 1 2 tan2 A 1 2 tan A tan2 2 tan A 1 tan A(1 2 A) 1 2 tan2 A 2 2 tan2 A 3 tan A 2 tan3 A 1 2 3 tan2 A ✔ 2. 1 5. 4 8. 2 y O c. a A !3B n 2 1 or a 3 Cumulative Review (pages 515–517) Part I 1. 4 4. 3 7. 1 10. 3 2 14. center: (1, 2) tan 2A 1 tan A 1 2 tan 2A tan A 2 tan A 1 2 tan2 A 1 tan A A 1 2 1 22tan tan2 A tan A A A 1 22tan tan2 A 1 tan A B 1 3. 2 6. 1 9. 3 342 8 8 n51 n51 n21 2 3 4 14580AK05.pgs 3/26/09 12:07 PM Page 343 Chapter 13.Trigonometric Equations 13-1 First Degree Trigonometric Equations (pages 524–526) 13-2 Using Factoring to Solve Trigonometric Equations (pages 529–530) Writing About Mathematics 1. The second equation simplifies to sin x 5 2 and 2 is outside the range of the sine function. 2. The second equation simplifies to tan x 5 1. Since the tangent function is periodic, there are an infinite number of x-values where tan x 5 1. Developing Skills 3. 60°, 300° 4. 150°, 330° 5. 90° 6. 90°, 270° 7. 135°, 315° 8. 45°, 315° 9. p3 , 2p 10. 0 3 5p 7p 11. p4 , 5p 12. 4 4, 4 p 3p 13. p6 , 5p 14. , 2 2 6 15. 49° 16. 79° 17. 71° 18. 24° 19. 12° 20. 18° 21. 75.5°, 284.5° 22. 16.6°, 163.4° 23. 104.0°, 284.0° 24. 131.8°, 228.2° 4p 25. 0.17, 2.97 26. 2.09, 4.19 or 2p 3, 3 27. 0.38, 3.52 28. 3.02, 6.16 Applying Skills 29. a. E 15 10 5 25 210 215 Writing About Mathematics 1. No. The method of using factoring to solve a trigonometric equation depends on the multiplicative property of zero: If ab 5 0, then a 5 0 or b 5 0. Thus, the right side of the equation must equal 0. 2. Yes. 2(sin u)(cos u) 1 sin u 1 2 cos u 1 1 5 0 sin u (2 cos u 1 1) 1 1(2 cos u 1 1) 5 0 (sin u 1 1)(2 cos u 1 1) 5 0 sin u 1 1 5 0 Developing Skills 3. 30°, 150°, 270° 4. 5. 60°, 120°, 240°, 300° 6. 7. 60°, 300° 8. 9. 45°, 63.4°, 225°, 243.4° 10. 11. 19.5°, 41.8°, 138.2°, 160.5° 12. 66.4°, 113.6°, 246.4°, 293.6° 13. 63.4°, 99.5°, 243.4°, 279.5° 14. 70.5°, 75.5°, 284.5°, 289.5° 15. 1.11, 1.25, 4.25, 4.39 16. 17. 0, p (3.14), 3.55, 5.87 18. 19. 0.25, 0.52, 2.62, 2.89 20. 21. 308 or p6 22. t O 1 30°, 150°, 210°, 330° 45°, 135°, 225°, 315° 90°, 210°, 270°, 330° 0°, 70.5°, 289.5° 0, 1.82, 4.46 3.48, 5.94 0.17, 1.11, 3.31, 4.25 2.03 23. 290°, 270° 2 13-3 Using the Quadratic Formula to Solve Trigonometric Equations (page 534) b. 220 volts c. 2 d. (1) u 5 0.93, 5.36 (2) t 5 0.30 s and 1.70 s 30. a. x 5 36.9° b. u 5 18.4° 1 31. a. We used the expression tan u for the cotangent function, which is undefined at p2 . b. Yes. cot A p2 ) 5 2 cos u 1 1 5 0 cos p 2 sin p 2 Writing About Mathematics 1. When the discriminant is negative, the solutions are imaginary numbers. 2. When factored, csc u 5 0 and csc u 5 12. The range of the cosecant function is (2`, 21g < f1, `) . Developing Skills 3. 202°, 338° 4. 74°, 125°, 254°, 305° 5. 29°, 99°, 261°, 331° 6. 14°, 166°, 246°, 294° 7. 111°, 159°, 291°, 339° 8. 46°, 80°, 280°, 314° 9. 50°, 157°, 230°, 337° 10. 72°, 144°, 216°, 288° 11. 55°, 125° 12. 39°, 119°, 219°, 299° 13. 64°, 140°, 220°, 296° 14. { } 5p 7p 15. p4 , 3p 16. 0.56, 5.72 , , 4 4 4 5 01 5 0 and sin A p2 2 p2 B 5 0. 343 14580AK05.pgs 3/26/09 12:07 PM Page 344 13-4 Using Substitution to Solve Trigonometric Equations Involving More Than One Function (pages 537–538) y Writing About Mathematics 1. Yes. The maximum value of both sin u and cos u is 1, and they are never equal to 1 for the same value of u. Therefore, their sum will always be less than 2. 2. 0° # u # 180°. Sine is positive in the first and second quadrants. Developing Skills 3. 30°, 150° 4. 30°, 150° 5. 45°, 90°, 225°, 270° 6. 30°, 150°, 270° 7. 0°, 60°, 300° 8. 30°, 150°, 210°, 330° 9. 45°, 135°, 225°, 315° 10. 90°, 270° 11. 45°, 135°, 225°, 315° 12. 0°, 180°, 210°, 330° 13. 30°, 150° 14. 45°, 135°, 225°, 315° Applying Skills 15. a. (1) 2!3 2 2 in. (2) 2 !2 2 2 in. b. 0° c. 628.96° 7y 7 cos u 5 1.75 2 5 8 u 5 arccos 78 5 28.968 Review Exercises (page 543) 1. 3. 5. 7. 9. 10. 11. 13. 15. 17. 18. 120°, 240° 2. 240°, 300° 60°, 300° 4. 60°, 180°, 300° 0°, 180, 360° 6. 60°, 120°, 240°, 300° 45°, 135°, 225°, 315° 8. 45°, 135°, 225°, 315° 22.5°, 202.5° 30°, 90°, 150°, 210°, 270°, 330° 3.43, 5.99 12. { } p , 3.39, 6.03 14. 0, 2.30, 3.98 2 1.34, 2.91, 4.48, 6.05 16. 1.20, 1.43, 4.85, 5.08 0, 0.62, 2.53, p, 3.76, 5.67 3p p 19. 1.33, 4.47 2 , 3.39, 2 , 6.03 4p 20. 1.23, 2p 3 , 3 , 5.05 22. p3 , 5p 3 13-5 Using Substitution to Solve Trigonometric Equations Involving Different Angle Measures (pages 540–541) 21. 1.11, 1.77, 4.25, 4.91 23. The left side of the equation is equal to zero only at values of u for which both the tangent and secant functions are undefined. sin u 1 tan u 2 sec u 5 cos u 2 cos u 5 0 sin u 5 1 u 5 p2 Writing About Mathematics 1. No. Dividing by 2 divides the coefficients, not the angles. 2. No. You must account for the factor cos u. The solution set also includes the numbers Up2 , 3p 2 V. Developing Skills 3. 30°, 90°, 150°, 270° 4. 0°, 180°, 360° 5. 0°, 180°, 360° 6. 30°, 90°, 150°, 210°, 270°, 330° 7. 30°, 150°, 210°, 330° 8. 60°, 300° 9. 30°, 90°, 150° 10. 45°, 135°, 225°, 315° 7p 9p 11. p5 , 3p 5, 5, 5 13. 0.34, p2 , 2.80 1.75y 20. a. sin u 5 x, sin 2u 5 x 5 4x b. sin 2u 5 1.75 sin u 2 sin u cos u 5 1.75 sin u sin u (2 cos u 2 1.75) 5 0, reject sin u 5 0 However, tangent is undefined at u 5 p2 . CD 24. a. tan u 5 AD CD tan 2u 5 DB b. CD 5 !5 5 AD CD c. tan u 5 AD 5 !5 5 d. u 5 arctan !5 5 5 24.018 e. m/A 5 248, m/B 5 488, m/C 5 1088 12. 0, 1.82, p, 4.46, 2p 14. 0.17, p2 , 2.97, 3p 2 15. 0, 1.15, 1.99, p, 4.29, 5.13, 2p 16. 1.36, 4.92 17. 0.28, 2.86 18. 0.12, 3.02 Applying Skills 19. a. d 5 90 sin u b. d 5 60 sin (908 2 u) c. 90 sin u 5 60 sin (908 2 u) Exploration (page 544) (1) a. A 5 4 tan u (2) a. A 5 2 tan u (3) a. A 5 12 sin 2u 5 sin u cos u b. b. b. (4) a. A 5 12 sin u (5) a. Area of triangle b. 0.12 0.24 p 4 p 2 5 12bh sin u 5 23 cos u tan u 5 23 5 12 sin (p 2 u) cos (p 2 u) 5 212 sin u cos u Area of semicircle 5 p2 u 5 arctan 23 5 33.698 d. 49.92 m Total area 5 p2 2 12 sin u cos u 344 14580AK05.pgs 3/26/09 12:07 PM Page 345 b. No possible value for u. Since the area of the semicircle circle is p2 < 1.57 square units, the minimum area of the shaded region is p2 . 1 square units. (6) a. A 5 2 sin u cos u 5 sin 2u p b. 12 Part III Cumulative Review (pages 545–546) (x 2 1)2 1 (y 2 2)2 5 72 Part IV Part I 1. 3 4. 1 7. 4 10. 1 Part II 11. 2. 1 5. 2 8. 4 3 3 3 3 13. 2, 2!9, 2!81, 18, 18!9, 18!81 1 14. Center 5 A 22 21 4, 5 2 2 B 5 (1, 2) Radius 5 " (22 2 4) 2 1 (5 1 1) 2 5 !36 1 36 5 6!2 15. sin u 5 13 3. 4 6. 3 9. 2 tan u 5 2 !2 4 16. y 1 21 O sec u 5 23 !2 4 5 6 2 2 5 2 x x 22 5 52 2 x6 x 22 5 5 22 6x x cot u 5 22 !2 0 5 2x2 2 6x 1 5 5 0 x x5 x5 12. cos u 5 22 !2 3 6 6 !36 2 4(2)(5) 4 3 1 6 i 2 2 x 5 4y 2 2 x 1 2 5 4y x 1 2 5y 4 2 21 f (x) 5 x 1 4 Chapter 14.Trigonometric Applications 14-1 Similar Triangles (pages 551–552) 15. 16. 17. 18. 19. 20. 21. 22. 23. Writing About Mathematics 1. Yes, since tan u 5 28.48 25.30 5 1.6, u can be found by using arctan. 2. Quadrant III. Both cosine and sine are negative when evaluated. Developing Skills 3. A2 !2, 2 !2B 4. A !3, 1B 5. (0, 6) 6. A24, 4 !3B 7. 8. (20.5, 0) 9. A 2152 !2, 152 !2 B A 292 !3, 92 B 11. (0, 212) 3 13. Q !3 2 , 22 R 10. A 2252 !3, 12. (21, 21) a. a. a. a. a. a. a. a. a. 10 13 7 15 15 14.42 25 11.66 11.31 24. a. R(5, 0), SQ1.5, 3 !3 2 R 25. a. R(12, 0), S(0, 8) 225 2 B 26. a. R(8, 0), SQ24!2, 4!2R 27. a. R(20, 0), S(10, 10) 28. a. R(9, 0), SA4.5, 4.5!3B 29. a. R(7, 0), SA8 !3, 8B 14. A1, 2!3B 345 b. b. b. b. b. b. b. b. b. 53° 113° 90° 323° 0° 236° 16° 301° 135° b. 154!3 sq units b. 48 sq units b. 16 !2 sq units b. 100 sq units b. 20.25 !3 sq units b. 28 sq units 14580AK05.pgs 3/26/09 12:07 PM Page 346 7. cos A 5 78 cos B 5 11 16 14-2 Law of Cosines (pages 555–556) Writing About Mathematics 1. Let C be the obtuse angle of ABC. By the Law of Cosines, c2 5 a2 1 b2 2 2ab cos C. Since the cosine of an obtuse angle is negative, 2(2ab cos C) is positive. Therefore, c2 5 a2 1 b2 1 2ab cos C. A whole is greater than the sum of its parts, so c2 . a2 or c . a, and c2 . b2 or c . b. 2. The cosine of a right angle is zero. Thus, when c is the hypotenuse, c2 5 a2 1 b2 2 2ab cos C 5 a2 1 b2, which is the Pythagorean theorem. Developing Skills 3. m2 5 a2 1 r2 2 2ar cos M 4. p2 5 n2 1 o2 2 2no cos P 5. 2 !7 6. !37 7. 8 !2 8. 4 9. !26 10. !13 11. 2 !19 12. 9 !7 13. 2 !10 14. 5.6 15. 147.0 16. 4.8 17. 98.6 18. 1.7 19. 7.5 Applying Skills 20. a. 0.72 mi b. 1.70 mi 21. 24.08 lb 22. a. 87 m b. 74 m 23. 28 ft 24. 151.1 m 25. 36.5 nautical miles 26. c2 5 2x2 2 2x2 cos 60° 5 2x2 2 x2 5 x2 6c5x cos E 5 29 48 cos F 5 61 72 13. 15. 17. 19. 21. 22. 24. 5. 20.575 4. cos Q 5 95 cos N 5 256 5 cos B 5 13 cos P 5 139 160 cos C 5 0 33°, 64°, 83° 14. 36°, 40°, 104° 42°, 51°, 87° 16. 47°, 47°, 86° 48°, 63°, 69° 18. 16°, 74°, 90° 37° 20. 122° a. 33.7 in. b. 58° 83° 23. 82° Let x 5 the length of any side of the equilateral triangle. Let C 5 any angle of the triangle. x2 cos /C 5 2x 2 cos /C 5 12 /C 5 608 14-4 Area of a Triangle (pages 563–564) Writing About Mathematics 1. Since /A and /B are supplementary, sin A 5 sin (180° 2 A) 5 sin B. 2. Yes. The area of the rhombus is (PQ)(PS)(sin P). Since the sides are congruent, (PQ)(PS)(sin P) 5 (PQ)2(sin P). Developing Skills 3. 3 sq units 4. 30 sq units 5. 60 sq units 6. 108 sq units 7. 16.8 sq units 8. 12 sq units 9. 77.5 sq units 10. 24,338.5 sq units 11. 12.6 sq units 12. 25,221.0 sq units 13. 122.0 sq units 14. 36,615.3 sq units 15. 400!3 m2 16. 36!2 cm2 17. 480 ft2 Applying Skills No C exists such that cos C is less than 21. c2 5 a2 1 b2 2 2ab cos C 2 c 1 2ab cos C 5 a2 1 b2 If /C is obtuse, 2ab cos C is negative; thus c2 . a2 1 b2. Developing Skills 1 2 2uv 12. cos A 5 12 13 2 x2 2 x2 cos /C 5 x 1 2x 2 2. 3. cos T 5 5 cos R 5 216 11. cos M 5 11 80 122 5 42 1 72 2 2(4)(7) cos /C 79 5 256 cos /C cos /C 5 279 56 , 21 t2 10. cos P 5 37 40 cos Q 5 13 20 17 9. cos D 5 2192 Writing About Mathematics 1. Let C be the angle opposite the side of length 12. Then: v2 cos B 5 34 cos C 5 34 cos C 5 214 14-3 Using the Law of Cosines to Find Angle Measure (pages 558–559) u2 8. cos A 5 218 18. a. 13 c. 4!2 km2 p2 1 r2 2 q2 2pr 6. 0 19. 234 ft2 346 b. 2 !2 3 d. 6 km2 20. 125!3 ft2 14580AK05.pgs 3/26/09 12:07 PM Page 347 8. a. 1 b. {3°, 27°, 150°} 9. a. 0 10. a. 2 b. {15°, 20°, 145°}, {5°, 15°, 160°} 11. a. 0 12. a. 1 b. {135°, 30°, 15°} 13. a. 1 b. {30°, 60°, 90°} 14. a. 2 b. {45°, 62°, 73°}, {45°, 17°, 118°} Applying Skills 15. a. 60.07° b. No, the triangle formed by the ladder, wall, and ground is a right triangle. 16. Yes, there can be only one garden. Angles: {37°, 68°, 75°} Sides: {5 ft, 7.7 ft, 8 ft} 17. No. Since 10 , 12, 10 cm must be the length of the short diagonal. Therefore, the other angle measures 60°. Using the Law of Sines to find the angle opposite the 12 cm side yields a value of sine greater than 1. 18. Yes. Two triangles are possible. Sides of 1st triangle: {2.0 km, 2.5 km, 2.7 km} Sides of 2nd triangle: {2.0 km, 2.5 km, 0.8 km} The route corresponding to the first triangle is longer. 21. sin 30° 5 sin (180° 2 30°) 5 sin 150° Thus, 21 (AB)(BC) sin 308 5 12 (DE)(EF) sin 1508. 22. a. 23 b. 41.8° or 138.2° c. Yes, an acute or an obtuse triangle. 23. a. A 5 ac sin u b. u 5 90° 14-5 Law of Sines (pages 567–568) Writing About Mathematics 1. No. A positive sine means the angle is in the first or third quadrant; that is, the angle can be either acute or obtuse. 2. Yes. A positive cosine means the angle is in the first or fourth quadrant. However, in a triangle, each angle when drawn in standard position is in the first or second quadrant. Thus, when the cosine is positive, the angle is acute. Similarly, a negative cosine means the angle is obtuse. Developing Skills 3. 6. 9. 12. 15. 16. 17. 18. 4. 48 3 !6 7. 12.5 4 !6 23.5 10. 31.4 18.3 13. 97.7 6.93 a. 8.85 cm b. 32.2 cm a. 31.1 in. b. 83 in. c a 5 sin A sin 908 a sin A 5 sin A 5 5. 8. 11. 14. 4!3 64 3 44.5 16.9 14-7 Solving Triangles (pages 579–580) Writing About Mathematics 1. Since BCD is a right triangle: c 1 a c BC 5 sin85358 < 148.193 ft Use the Law of Sines in BCA: Applying Skills 19. a. 3.18 ft b. 12.3 ft 20. 138.0 ft, 250.2 ft 21. a. 14.0 ft b. 18.6 ft 22. 3.1 mi 23. $5,909 BA sin 408 148.193 5 sin 1058 BA < 98.62 ft 2. An angle of depression is the complement of the complement of the angle of elevation. Taking the complement of a complement is congruent to the original angle. Developing Skills 3. a. Law of Cosines b. 4.9 4. a. Law of Sines b. 59.0°, 121.0° 5. a. Law of Cosines b. 11.6° 6. a. Law of Cosines b. 76.9° 7. a. Both b. 115.7° 8. a. Law of Cosines b. 122.6° 9. a. Law of Sines b. 12.7° 10. a. Law of Sines b. 8.2 11. c 5 17, /A 5 518, /B 5 698 12. c 5 20, /A 5 258, /C 5 1258 13. a 5 34, c 5 31, /A 5 758 14. b 5 5, c 5 7, /A 5 908 15. f 5 99, /D 5 438, /E 5 278 14-6 The Ambiguous Case (pages 573–574) Writing About Mathematics 1. Yes, the side opposite the largest angle, 110°, has to be the largest side of the triangle. Since it is not, no triangle can exist with these measurements. 2. Because we know that one of the angles is obtuse, we cannot use the ambiguity of the sine function to imply two possible triangles. Developing Skills 3. a. 2 b. {20°, 155°, 5°}, {20°, 25°, 135°} 4. a. 1 b. {30°, 60°, 90°} 5. a. 1 b. {39°, 49°, 92°} 6. a. 0 7. a. 1 b. {29°, 31°, 120°} 347 14580AK05.pgs 3/26/09 12:07 PM Page 348 16. q 5 13, r 5 6, /P 5 708 17. /R 5 1098, /S 5 448, /T 5 278 18. C 5 36, /A 5 288, /B 5 228 19. /P 5 378, /Q 5 538, /R 5 908 20. f 5 62, /E 5 908, /F 5 608 21. t 5 23, /R 5 408, /S 5 508 22. /A 5 258, /B 5 1358, /C 5 208 23. 27.4 Applying Skills 24. a. 78.6°, 101.4° b. 37.8°, 40.8°, 60.6° c. 2.3 km 25. 107 ft 26. 16.6 ft, 19.2 ft 27. 35 ft 28. 55.6°, 71.4°, 116.5°, 116.5° Cumulative Review (pages 585–586) Part I 1. 2 2. 1 3. 4 4. 3 5. 1 6. 2 7. 3 8. 3 9. 3 10. 3 Part II 11. cos 158 5 cos (458 2 308) 5 cos 458 cos 308 1 sin 458 sin 308 !3 !2 1 5 !2 2 ? 2 1 2 ?2 !2 5 !6 1 4 12. 5 Review Exercises (pages 582–584) 5 1. 21 2. 17.5 in. 3. 50° 4. 112° 2 1 242 2 262 50 5. cos u 5 10 2(10)(24) 6 u 5 90 + 6. a. 240 sq units b. 20 7. a. 31.3° b. 42.8 sq units 8. AB = AC 5 26.2 9. 15, 21 10. a. 2 b. /B 5 588, /C 5 748 c. /B 5 1228, /C 5 108 P 668 11. sin R 5 r sin 5 15 sin 5 1.14 . 1 p 12 5 1 2 !5 1 2 !5 ? 1 1 !5 1 2 !5 1 2 2 !5 1 5 1 2 5 6 2 2 !5 24 !5 2 3 2 Part III 13. Answer: x 5 2 3 2 !2x 2 3 5 x 3 2 x 5 !2x 2 3 x2 2 6x 1 9 5 2x 2 3 x2 2 8x 1 12 5 0 (x 2 6)(x 2 2) 5 0 x56 x52 Reject extraneous root 14. Answer: {210°, 330°} 6 sin2 x 2 5 sin x 2 4 5 0 A x 1 12 B A x 2 43 B 5 0 12. a. 53°, 127° b. 25.0 in. c. Using the answers to part a: 626 in.2 13. 49 5 64 1 c2 2 (16)(c) cos 608 0 5 c2 2 8c 1 15 c 5 3, 5 sin x 5 212 A 5 8 sin7 608 5 4 !3 14. sin B 5 b sin a 7 m/B 5 81.88 m/B 5 1808 2 81.88 5 98.28 15. 25.4 ft 16. From A: 8.7 mi, from B: 7.0 mi 17. a. 24 b. 37° 25 sin x 5 43 x 5 2108, 3308 Part IV ?4 15. logb x 5 logb A 3!8 2 x5 3(16) 2 !2 Reject extraneous root B x 5 12!2 16. a–b. y c. Using the answer to part b: 13.9 Exploration (pages 584–585) c. 2 1 Part A Answers will vary. DEF is an equilateral triangle. Part B Steps 1–8. Answers will vary. Step 9. Yes, DEF is an equilateral triangle. O 348 p 2 x 14580AK05.pgs 3/26/09 12:07 PM Page 349 Chapter 15. Statistics 15-1 Gathering Data (pages 594–595) 7. Writing About Mathematics 1. A control group is necessary to ensure that any changes to members of the experimental group are due to the medication and not to some external factor, such as the placebo effect. 2. It is necessary that a participant does not know to which group he or she belongs because this knowledge can influence the participant’s perception of the effectiveness of the treatment. Developing Skills 3. Stem Leaf 9 0025 8 2455678 7 4568 6 6778 5 4 8. 10. Frequency Leaf 1 01223447 0112246777789 799 Key: 12 9 129 No. of Books Read 8–9 6–7 4–5 2–3 0–1 5–9 10–14 15–19 20–24 25–29 30–34 35–39 14 12 10 8 6 4 2 0 Frequency 1 3 5 9 7 0 –5 41 0 –6 51 0 –7 61 0 –8 71 0 –9 81 00 –1 91 0 11 1– 10 xi 11. Frequency 6. 18 16 14 12 10 8 6 4 2 0 xi Key: 15 5 155 5. Stem 15 14 13 12 Frequency 1 2 11 16 20 Frequency Leaf 6 4 0 1357 025566 2478 357 9 5 Frequency 3 11 7 5 No. of Siblings 6–7 4–5 2–3 0–1 9. Key: 5 4 54 4. Stem 23 22 21 20 19 18 17 16 15 Size 15–17 12–14 9–11 6–8 20 16 12 8 4 0 25–29 30–34 35–39 40–44 45–49 50–54 55–59 xi 349 14580AK05.pgs 3/26/09 12:07 PM Page 350 15-2 Measures of Central Tendency (pages 604–605) Applying Skills In 12–18, answers will vary. 12. Record a sample of the temperature twice daily, at perceived high and low temperatures, and take the average over each month. 13. Conduct a survey on a sample, such as every tenth person leaving the restaurant on a given day. 14. Conduct an observational study on a sample, such as recording the patient’s temperature every 2 hours. 15. Conduct a census on the population, recording all students’ grades on the test. 16. Conduct a census on the population, counting the number of people living in each house, apartment, etc. (Note that because of the size of the population, methods involving random samples will need to be used.) 17. Conduct a survey on a sample, such as measuring the height of every fifth student enrolling in kindergarten in each elementary school. 18. Conduct a survey on a sample population of moviegoers, such as questioning every tenth person leaving 100 randomly selected theaters where the movie is showing. 19. a. Stem Leaf 9 23588 8 2344667789 7 245677 6 16 5 38 Writing About Mathematics 1. No. Whenever the number of data values of a set is odd, the number of data values less than the lower quartile or greater than the upper quartile cannot total exactly 50% of the number of data values. 2. Yes. Whether a set has 2n or 2n 1 1 data values, there are n data values above the median and n data values below the median. Developing Skills Mean 5 81.1, median 5 80, mode 5 80 Mean 5 66, median 5 65.5, mode 5 68 Mean 5 122.4, median 5 117, modes 5 115, 118 Mean 5 2.4, median 5 2, modes 5 0, 2 Mean 5 $8.26, median 5 $7.88, mode 5 $7.50 Mean 5 $3.48, median 5 $3.50, mode 5 $5.00 Q1 5 6.5, median 5 15, Q3 5 21 Q1 5 36, median 5 42.5, Q3 5 44 Q1 5 19, median 5 26, Q3 5 28.5 Q1 5 81, median 5 87, Q3 5 90.5 Q1 5 58, median 5 62, Q3 5 66 Q1 5 19.5, median 5 26, Q3 5 30 a. 90 b. 92 or 95 c. Any number other than 90, 92, or 95 16. Q1 5 25.5, Q2 5 50.5, Q3 5 75.5 17. Q1 5 24.5, Q2 5 50, Q3 5 75.5 Applying Skills 18. a. 79.52 b. 82 c. Q1 5 74, Q3 5 89 d. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. Key: 5 3 53 b. Score 91–100 81–90 71–80 61–70 51–60 Frequency 5 10 6 2 2 c. 21 d. 2 20. a. 2 b. 3 d. 72 e. 10 Hands-On Activity Answers will vary. 74 47 19. a. 200.7 b. 202.5 d. Q1 5 191.5, Q3 5 208.5 e. 82 89 99 c. 202 c. 12 178 20. 83 Hands-On Activity Answers will vary. 350 191.5 202.5 208.5 223 14580AK05.pgs 3/26/09 12:07 PM Page 351 15-3 Measures of Central Tendency for Grouped Data (pages 611–614) 5. Range 5 81 Interquartile range 5 53 6. Range 5 40 Interquartile range 5 4 7. Mean 5 35, median 5 35, range 5 30, interquartile range 5 10 8. Mean 5 7, median 5 7, range 5 8, interquartile range 5 2 9. Mean 5 24.9, median 5 25, range 5 26, interquartile range 5 17.5 Applying Skills 10. a. Mean 5 2,296.5, median 5 480 b. The median is more representative. The mean is strongly influenced by the outlier. c. The outlier is 19,014. d. With outlier removed, mean 5 439 and median 5 427. e. The mean is more representative of the data with the outlier removed. 11. a. 75.6 b. 80 c. Q1 5 65, Q3 5 88.5 d. 53 e. 23.5 12. Range 5 4, interquartile range 5 2 13. Range 5 9, interquartile range 5 2.5 14. a. 14 b. 3 c. 15 15. Range 5 1, interquartile range 5 0.4 16. a. Range of A 5 9 5 range of B 5 9; yes, they are the same. b. Post Office A: Writing About Mathematics 1. Not necessarily. If the ages are not distributed perfectly evenly, then Adelaide cannot make this assumption. 2. Yes. There are only six possible ages those employees could be, so there must be some employees with the same age. Developing Skills 3. Mean 5 3.08, median 5 3, mode 5 3 4. Mean 5 32.4, median 5 30, mode 5 30 5. Mean 5 8.84, median 5 9, mode 5 9 6. Mean 5 6.63, median 5 7, mode 5 7 7. Mean 5 $1.34, median 5 $1.30, mode 5 $1.30 8. Mean 5 81.6, median 5 80, mode 5 85 9. 19th percentile 10. 16th percentile 11. 36th percentile 12. 32nd percentile 13. Mean 5 12.8, median 5 12.6 14. Mean 5 78.5, median 5 78.5 15. Mean 5 $1.3, median 5 $1.3 16. Mean 5 11.3, median 5 10.9 17. Mean 5 $33.9, median 5 $27.50 18. Mean 5 0.2, median 5 0.2 Applying Skills 19. Mean 5 11.625, median 5 12, mode 5 12 20. Mean 5 17.43, median 5 17 21. a. Mean 5 35.45, median 5 35 b. 25th percentile 22. Mean 5 251.875, median 5 253.58 23. Mean 5 48.8, median 5 51.125 Hands-On Activity Answers will vary. * * 1 2 3 9 10 9 10 Post Office B: 15-4 Measures of Dispersion (pages 617–619) Writing About Mathematics 1. No. The subscript for each data value indicates its position in a list of data values, not its value. 2. Yes. An outlier is a data value that is 1.5 times the interquartile range below the first quartile or above the third quartile. For the given information, the interquartile range is 6, and 12 2 (1.5)(6) 5 3, which makes the data value 2 an outlier. Developing Skills 3. Range 5 16 Interquartile range 5 10 4. Range 5 22 Interquartile range 5 8 1 2 3 4 5 6 7 8 c. Interquartile range of A 5 1, interquartile range of B 5 5 d. Post Office A. Wait times there of 9 and 10 minutes are outliers, which is not the case at Post Office B. Hands-On Activity Answers will vary. 351 14580AK05.pgs 3/26/09 12:07 PM Page 352 15-6 Normal Distribution (pages 632–634) 15-5 Variance and Standard Deviation (pages 625–627) Writing About Mathematics 1. The mean of these scores is 90. In a normal distribution, 50% of the scores are below the mean. Only one of these five scores is below 90. 2. No. In a normal distribution the intervals closest to the mean contain more of the scores. Scores are not uniformly distributed through the first standard deviation above the mean. Developing Skills 3. 68% 4. 81.5% 5. 81.5% 6. 84% 7. 84% 8. 50% 9. 50% 10. a. 45 b. 52 c. 35 d. 28 Applying Skills 11. (4) 12. (2) 13. (3) 14. (3) 15. a. About 0.62%. Use normalcdf(0, 16, 16.1, 0.04). b. About 98.8%. Use normalcdf (16, 16.2, 16.1, 0.04). 16. About 1.3% of the time. Ken can expect to be punctual approximately 98.7% of the time. This means he will be late approximately 100% 2 98.7% 5 1.3% of the time. 17. Approximately 91.04% of patrons check out fewer than 7 books. 18. 20.5 19. 8 20. The science test. On the math test, Nora’s score was within 2 standard deviations of the mean. On the science test, her score was more than 3 standard deviations above the mean. Writing About Mathematics 1. The second data set (from the sample) has the larger standard deviation since its denominator is smaller. 2. Yes. If the standard deviation is the square root of the variance, then the variance is the square of the standard deviation. Developing Skills 3. Variance 3.92 Standard deviation 1.98 4. Variance 8.29 Standard deviation 2.88 5. Variance 116.74 Standard deviation 10.80 6. Variance 1,223.14 Standard deviation 34.97 7. Variance 66.24 Standard deviation 8.14 8. Variance 6.65 Standard deviation 2.58 9. Variance 233.36 Standard deviation 15.28 10. Variance 32.99 Standard deviation 5.74 11. Variance 877.38 Standard deviation 29.62 12. Variance 12.57 Standard deviation 3.55 13. Variance 648.99 Standard deviation 25.48 14. Variance 106.78 Standard deviation 10.33 15. Variance 3.23 Standard deviation 1.80 16. Variance 4.20 Standard deviation 2.05 Applying Skills 17. Line A. Since its standard deviation is smaller, its late times are more closely clustered around the mean of 10 minutes. 18. a. Variance 269.43 b. Standard deviation 16.41 19. Variance 0.66; standard deviation 0.81 20. 2.21 21. 2.14 22. 5.93 Hands-On Activity Answers will vary. 15-7 Bivariate Statistics (pages 638–640) Writing About Mathematics 1. Univariate data consists of one number for each data point, or a single set of numbers. Bivariate data consists of two numbers for each data point, or two different sets of numbers. Example answers will vary. 2. A positive slope reflects a positive correlation and a negative slope reflects a negative correlation. Slope cannot be used to measure the strength of a correlation. Developing Skills 3. Bivariate 4. Univariate 5. Bivariate 6. Univariate 7. Moderate linear correlation 8. No linear correlation 352 14580AK05.pgs 3/26/09 12:07 PM Page 353 14. a. 7 6 5 4 3 2 1 0 Mother’s family Miles 9. Strong linear correlation 10. No linear correlation Applying Skills 11. a. 330 300 270 240 210 190 160 130 0 2 3 4 5 6 Family b. No linear correlation 15. a. 4.0 5.0 6.0 7.0 8.0 9.0 10.0 11.0 Gallons 360 340 320 300 280 260 240 220 200 180 160 140 120 100 0 20 40 60 80 100 120 140 160 180 200 220 240 260 280 300 320 340 Sales Temperature b. Strong positive linear correlation c. y 5 30.714x 2 4.166 12. a. 70 65 60 55 50 45 40 35 30 25 20 10 12 14 16 18 20 22 24 26 28 30 32 Seconds Number of ads b. Strong positive linear correlation c. y 5 0.765x 1 99.480 16. a. 50 45 40 35 30 25 20 15 10 5 Fat % of speeding accidents b. Strong positive linear correlation c. y 5 2.365x 2 2.145 13. a. 16 14 12 10 8 6 4 2 0 1 200 220 240 260 280 300 320 340 360 0 Calories 15 20 25 30 35 40 45 50 55 60 65 Age b. Moderate positive linear correlation c. y 5 0.065x 2 7.681 b. Strong negative linear correlation c. y 5 20.965x 1 64.990 353 14580AK05.pgs 3/26/09 12:07 PM Page 354 17. a. 1,800 b. Close to 1. There appears to be a very strong positive linear correlation. c. r 5 0.99 16. a. 10 8 6 4 2 0 22 24 1,700 Pounds lost Score 1,600 1,500 1,400 1,300 1,200 0 1 140 150 160 170 180 190 200 210 6 7 8 9 10 11 Month Weight b. Close to 21. There appears to be a very strong negative linear correlation. c. r 5 20.96 17. a. 7 6 5 4 3 2 1 0 5 10 15 20 25 30 35 40 b. No linear correlation 15-8 Correlation Coefficient (pages 645–646) High paying jobs Writing About Mathematics 1. No. When r 5 1 there is a perfect linear relationship between the data values, while a correlation coefficient of 0 indicates no linear relationship exists between the data values. 2. 1. There is a perfect linear relationship between temperature measured in degrees Fahrenheit and measured in degrees Celsius (otherwise they wouldn’t be measuring the same thing!). Developing Skills 3. 1 4. 0 5. 21 6. 0 7. Strong positive 8. Strong negative 9. None 10. Moderate/weak positive 11. Strong positive 12. Moderate negative 13. None 14. Strong negative Applying Skills 15. a. 550 500 450 400 350 300 250 200 150 100 Total jobs Actual temperature b. Close to 0. There does not appear to be a strong correlation. c. r 5 0.38 18. a. 70 65 60 55 50 45 40 35 Miles 2 4 2 3 4 5 45 50 55 60 65 70 Same-day forecast b. Closer to 1. There appears to be a moderate positive linear correlation. c. r 5 0.75 6 8 10 12 14 16 Gallons 354 14580AK05.pgs 3/26/09 Actual temperature 19. a. 12:07 PM Page 355 10. a. 90 85 80 75 70 65 60 55 50 45 40 0 45 50 55 60 65 70 2 3 4 5 6 b. Exponential. The scatter plot resembles an exponential curve. The curve does not pass through the origin, yi . 0, and the y-intercept is positive. c. y 5 8.609(1.560)x b. Closer to 1. There appears to be a moderate positive linear correlation. c. r 5 0.59 20. a. 1. There would be a perfect positive linear correlation. b. Greater than. Yes. Same-day forecasts should be more accurate than forecasts for five days in the future. 11. a. 15-9 Non-Linear Regression (pages 651–654) Writing About Mathematics 1. The function y 5 ln x is undefined for x 5 0. 2. Function y 5 axb has only positive or only negative y-values when b is even. If b is odd, the power function will have both positive and negative y-values. Developing Skills 3. Quadratic 4. Exponential 5. Logarithmic 6. Exponential 7. Power 8. Cubic y 1 x Five-day forecast 9. a. 100 90 80 70 60 y 50 40 30 20 10 0 23 22 21 0 18 15 12 9 6 3 0 y 23 26 29 212 215 218 221 24 23 22 21 0 1 2 3 x b. Cubic. The scatter plot resembles a cubic curve. c. y 5 0.291x3 2 0.027x2 1 0.557x 1 0.467 16 14 12 10 8 6 4 2 0 0 1 2 3 4 5 6 7 8 9 10 x b. Quadratic. The scatter plot appears to be quadratic. c. y 5 0.968x2 2 11.705x 1 40.950 355 4 14580AK05.pgs 12. a. 3/26/09 12:07 PM Page 356 16. a. The power regression equation, y 5 123,113.744x–1.981. b. Yes. When Neptune’s orbital speed is plugged into the regression equation, we get 4,277.2 million km as its distance from the sun, which is a reasonably good estimate. 17. a. 275 22 23 24 y 25 26 27 28 0 1 2 3 4 5 6 7 8 9 10 11 Volume (ft3) x b. Logarithmic. The scatter plot resembles a logarithmic curve that does not pass through the origin, xi . 0, and the y-intercept appears negative. c. y 5 26.995 1 2.003 ln x 2.9 2.7 2.5 2.3 2.1 1.9 y 1.7 1.5 1.3 1.1 0.9 0.7 0 1 2 3 4 5 6 Height (ft) b. Power regression. It resembles the positive half of a power function passing through (0, 0), xi . 0, and yi . 0. c. y 5 2.024x2.991 18. a. 0 1 2 3 4 5 6 7 8 9 Temperature (°F) 13. a. 250 225 200 175 150 125 100 75 50 25 0 10 11 x b. Exponential. The scatter plot resembles an exponential curve that does not pass through the origin, yi . 0, and the y-intercept is positive. c. y 5 3.127(0.859)x Applying Skills 14. a. y 5 999.843(1.045)x b. $1,623.00 15. a. y 5 19.051 1 5.074 ln x b. 37.5 in. 70 69 68 67 66 65 64 63 62 61 60 0 1 2 3 4 5 Setting b. y 5 60.811x0.076 356 6 14580AK05.pgs 3/26/09 12:07 PM Page 357 Developing Skills In 3–9, parts b and c, answers shown are obtained using the rounded regression equation from part a. 3. a. y 5 0.064x 1 0.971 b. 1.336 c. 4.359 4. a. y 5 0.486x 1 2.607 b. 8.439 c. 282.702 5. a. y 5 1.665x 2 1.147 b. 20.315 c. 0.989 6. a. Logarithmic: y 5 0.800 1 0.413 ln x b. 0.939 c. 5.446 7. a. Quadratic: y 5 0.989x2 2 39.930x 1 407.290 b. 70.546 c. 11.436, 28.938 8. a. Exponential: y 5 5.008(2.240)x b. 23,837.928 c. 3.713 9. a. Power: y 5 0.578x2.716 b. 493.156 c. 8.840 Applying Skills In 10–16, answers shown are obtained using the rounded regression equation. 10. The linear regression model is y 5 0.206x 2 400.971. a. 10.63% b. 14.12% 11. The linear regression model is y 5 0.041x 2 1.851. a. 5.53 sec b. 11.47 sec 12. The linear regression model is y 5 1.8x 1 32. a. 77°F b. 220°C 13. Let x 5 the number of years since 1980. The exponential model is y 5 78.753(1.187)x. a. 2,428 cars b. 56 cars 14. The power model is y 5 60.811x0.076. a. 65.2°F b. 4.35°F 15. The power model is y 5 2.000x3.000 or y 5 2x3. a. 3.91 ft3 b. 3.7 ft c. 7.4 ft 16. a. The logarithmic model is y 5 6.784 1 5.063 ln x. b. 10.3 in. c. 22.0 in. Speed 19. 3,500 3,000 2,500 2,000 1,500 1,000 500 0 0 3 6 9 12 15 18 21 24 27 30 33 36 Year Temp (°F) The exponential regression model is y 5 0.407(1.294)x. No, Moore’s Law does not appear to hold for Intel chips. According to this model, the speed of Intel computer chips increases by a factor of 1.2942 1.67 every two years. Hands-On Activity: Sine Regression a. 85 80 75 70 65 60 55 50 45 40 35 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 Month b. y 5 23.442 sin (0.527x 2 1.713) 1 62.481 c. 60° d. 72° Review Exercises (pages 664–668) 15-10 Interpolation and Extrapolation (pages 658–661) 1. Univariate 2. Bivariate 3. Bivariate 4. Census: counting data of general interest for an entire population. Survey: asking questions (oral or written) to find out experiences, preferences, or opinions. Controlled experiment: structured study, usually of two groups, to compare results of a treatment or other process that only one group undergoes. Observational study: structured study, usually of two groups, in which researchers do not impose the treatment on either group. Writing About Mathematics 1. Interpolation is estimating a function value between given values. Extrapolation is estimating a function value outside the range of given values. 2. A major source of error when using extrapolation is that the regression model does not always hold outside of the given range of values. 357 14580AK05.pgs 3/26/09 12:07 PM Page 358 13. a. 16 14 12 10 8 6 4 2 0 22 24 26 Gain or loss 5. a. A sample. The student did not obtain information for every 9th grade student in the state. b. No. The data collected cannot be expected to reflect the grades of all students taking the test. The sample was very small and was not representative of the population as a whole, since the data was gathered from only one high school in the state. 6. a. 28 b. 80 c. 81 d. Q1 5 79, Q3 5 86 e. 7 f. Yes. The grade of 60 is an outlier since it is less than 1.5 times the interquartile range below the lower quartile: 79 2 1.5(7) 5 68.5. 0 5 10 15 20 25 30 35 40 45 50 55 60 Stock price g. * 60 7. a. 38.2 d. 6 f. 2 8. 9. 10. 11. 12. 70 78 80 82 84 86 14. 15. 16. 17. 18. 20. 21. 88 b. 38 c. 38 e. Q1 5 37, Q3 5 39 g. 2.43 h. 1.56 i. The STAT menu on the calculator yields the same values as those found in parts a–h. The sample mean is 84.49 seconds. The sample variance is 1.957 seconds. a. 70% b. 96% c. The data appears to approximate a normal distribution. The data appears bell-shaped, 70% (close to the normal 68%) of the data is within one standard deviation of the mean, and 96% (close to the normal 95%) of the data is within 2 standard deviations of the mean. a. Moderate negative linear correlation b. Negative a. Strong positive linear correlation b. Positive a. Strong negative linear correlation b. Negative a. Moderate positive linear correlation b. Positive b. Yes, moderate positive linear correlation a. y 5 1.020x 1 0.024 b. r 5 0.999 a. y 5 4x 1 47.5 b. r 5 0.970 y 5 102.722(1.166)x y 5 699.397 2 250.239 ln x 24.4 million people 19. $67,500 351 dozen cookies a. 179 deer b. In the 7th year Exploration 1. y 5 13.619x2.122 2. y 5 35.938 1 1.627 ln x Cumulative Review (pages 669–671) Part I 1. 2 4. 1 7. 1 10. 4 Part II 11. x2 2 6x 1 13 x5 2. 3 5. 4 8. 2 3. 2 6. 4 9. 3 50 2b 6 "b2 2 4ac 2a 6 6 4i 2 x5 x 5 3 6 2i 12. y 1 22p 2p21 O p 22 358 x 2p 14580AK05.pgs 3/26/09 12:07 PM Page 359 Part III !6 9 3 13. log 5 1 3 m/BAT 5 180 2 50 5 1308 m/ATB 5 180 2 130 2 40 5 108 20 AT sin 108 5 sin 408 log 6 2 log 9 5 13 (log 2 1 log 3) 2 log 32 AT 5 74.033 TC sin 508 5 AT TC < 56.71 ft To the nearest foot, the height of the tree is 57 feet. 16. y 5 13 (log 2 1 log 3) 2 2 log 3 5 13 (a 1 b) 2 2b 5 13a 2 53b 14. 27x 1 1 5 81x (33) x 1 1 5 (34) x 33x 1 3 5 34x 3x 1 3 5 4x x53 Part IV 15. Let T 5 the top of the tree. Let C 5 the base of the tree. 1O 1 x Solutions: (0, 0) and (1, 1) Chapter 16. Probability and the Binomial Theorem 16-1 The Counting Principle (pages 675–678) 34. 35. 36. 37. 38. 39. 40. a. 72 b. 18 c. 36 256 a. 24 b. 12 a. 720 b. 120 a. 4,096 b. 4 a. 720 b. 240 a. 10,000 b. 5,040 c. Of the 10,000 telephone numbers with this prefix, 5,000 form an even number. 41. 79 42. 69 Writing About Mathematics 1. In the first situation, choosing a boy and choosing a girl are independent events. In the second situation, the choice of the first girl affects the choice of the second girl, and so the events are dependent. 2. The first is a dependent event, the second is an independent event. In the first situation, there are 52 3 51 5 2,652 possible outcomes. In the second, there are 52 3 52 5 2,704 possible outcomes. Developing Skills 3. 24 4. 120 5. 336 6. 132 7. 256 8. 625 9. 64 10. 125 11. 16 events 12. Independent 13. Dependent 14. Dependent 15. Independent 16. Independent 17. Dependent 18. 216 19. 32 20. 60 21. 48 22. 5,040 23. 30 24. 465 25. 3,993,600 26. 120 27. 12 Applying Skills 28. 6,720 29. 336 30. 15,120 31. 25 32. a. 462 b. 484 33. 756 16-2 Permutations and Combinations (pages 685–687) Writing About Mathematics P n! (n 2 r)! n! 5 (n 2 r)! 1. nCr 5 nr!r 5 3 r! r! 2. n! 5 n(n 2 1)(n 2 2) c1 5 n(n 2 1)! Developing Skills 3. 120 4. 479,001,600 5. 6,720 6. 9 7. 604,800 8. 720 9. 1,680 10. 720 11. 20 12. 4 13. 792 14. 792 15. 210 16. 3,003 17. 3,003 18. 1 19. 1 20. 120 21. 1 22. 1 23. 720 24. 180 25. 120 26. 3,360 27. 40,320 28. 37,800 29. 50,400 30. 4,989,600 359 14580AK05.pgs 3/26/09 12:07 PM Page 360 429 5 .3575 17. a. 1,200 b. No. The theoretical probability for rolling a 5 is 16 5 .16. The die may be rigged. 31. Order is not important and the chips are taken without replacement. a. 84 b. 1 c. 8,190 d. 10,080 e. 1,170 f. 3,240 g. 336 h. 756 32. 5,040 33. 15,120 34. 13,860 35. 1,814,400 36. 4,200 37. 720 38. 12 39. 10 40. 5 41. 10 Applying Skills 42. 15 18. 16% 19. a. Probability of exactly 2 plain: 10 C2 43. 12 C1 3 4C1 3 20C3 c. d. 22. b. 11. 12 180 5 13. a. 61 14. a. 53 12. 9! 2!2! 11! 2!2!2! 5 5. 36 5 3C1 3 15C2 315 18C3 48 C4 50C4 18 C5 50 C5 207 5 245 < 0.8449 153 5 37,835 < 0.0040 c. No, the empirical probability is much higher than the theoretical probability. This is likely because many players have some skill and therefore have a better than random chance of hitting the bull’s-eye. 25. 16 in.2 2 26. a. 56 ? 56 ? 16 5 563 < .1157 4 b. 56 ? 56 ? 56 ? 56 ? 16 5 565 < .0804 n21 c. 56n 2 5 16 10. 12 1 15 240 5 1,140 < .2105 A 1,954 1 b. 35,960 < .000028 1 8. 221 3 6C1 3 4C1 20C3 p 24. a. A bull 5 36 2 < 0.0024 board 72 b. 1,270 < 0.0567 1. a. Yes. 2,000 5 .977 5 97.7% b. Empirical, since it is based on real data. 2. Yes. Since the total probability of someone getting the part must equal 1, the probability of it being Casey is 0.4. Developing Skills 6. a. 35,960 10 C1 270 5 1,140 < .2368 1 3 14 2 315 5 45 5 0.4 Answer 120 6C3 5 4,896 < 0.0245 18 C3 23. a. Writing About Mathematics 4. 23 3 6 C1 20C3 240 5 1,140 < .2105 5 816 0.3860 b. There is 1 way to choose Stephanie. There are 14 pairs involving Jan. There are 3C1 3 15C2 5 315 possible choices. 21. a. 16-3 Probability (pages 691–694) 3. 16 10 C2 10C1 20. 99.6% 96! 12! b. (93!)3! ? (3!)(9!) 5 31,433,600 c. 924 56. a. 720 b. 120 57. 60,060 9. 450 cinnamon: 6 96! 55. a. (90!)6! 5 927,048,304 7. 10 C1 5 1,140 < .3947 b. Probability of exactly 1 maple, exactly 1 apple- 44. nCn 2 r 5 fn 2 (n 2n!r)g!(n 2 r)! 5 r!(n n!2 r)! 5 nCr 45. 56 46. 120 47. 18,876 48. 144 49. 10! 50. 2,598,960 4! 5 151,200 51. 116,396,280 52. 10 53. 3,024 54. a. 5,040 b. 17 c. 1 1 13 1 5,525 3 20 C3 d. From the answer in part c, we can see that the common ratio is 56 . 1 55 b. 41 c. 32 d. 21 1 b. 10 19 c. 20 7 d. 20 27. a. b. 20 C3 1,140 5 2,024 5 506 < .56 20 C2 3 4C1 24C3 4 28. a. 16 5 0.25 Applying Skills 15. .4 526 474 5 .526, tails 5 1,000 5 .474 16. Heads 5 1,000 4 b. 16 5 0.25 360 285 24 C3 760 95 5 2,024 5 253 < .38 14580AK05.pgs 3/26/09 12:07 PM Page 361 16-4 Probability with Two Outcomes (pages 699–700) Developing Skills Writing About Mathematics 1. 3,696 12 C2 3 8 C3 20 C5 5 15,504 < .2384. This is not a Bernoulli experiment because each student is chosen 7. 7C5 A 23 B 5 A 13 B 2 d. 609 625 15 15 r510 r510 7 7 r50 r50 10. .93 11. .49 12. .54 13. .84; for the upper limit, use any value more than 3 standard deviations above the mean. 14. .224 15. .453 16. .176 17. .045 c. 0.0017 f. No kings Applying Skills 19. .9998 20. .4315 21. a. .0460 22. .382 b. .1056 23. .141 24. .655 Writing About Mathematics 1. nCr 5 nCn–r 2. Yes, A x 1 x1 B 5 a nCi xn2i A x1 B n i50 0.20 0.15 0.10 0.05 0.00 7 8 9 n i50 Developing Skills 3. x6 1 6x5y 1 15x4y2 1 20x3y3 1 15x2y4 1 6xy5 1 y6 4. x7 1 7x6y 1 21x5y2 1 35x4y3 1 35x3y4 1 21x2y5 1 7xy6 1 y7 5. 1 1 5y 1 10y2 1 10y3 1 5y4 1 y5 6. x5 1 10x4 1 40x3 1 80x2 1 80x 1 32 7. a4 1 12a3 1 54a2 1 108a 1 81 8. 16 1 32a 1 24a2 1 8a3 1 a4 9. 8b3 2 12b2 1 6b 2 1 10. 24 1 4i 0.25 6 i 5 a nCi xn2ix2i a nCi xn22i. 0.30 5 n i50 n 0.35 4 18. 12 16-6 The Binomial Theorem (pages 710–711) 0.45 0.40 Probability 513 c. 625 r50 5 .2755 3 b. 297 625 3 Writing About Mathematics 1. No. Exactly r is included in both “at least” and “at most,” so their sum will be greater than 1. 2. No, as you can see from the histogram of the probabilities, the graph is not bell-shaped. 2 5. a. 35 r 102r 9. a 10Cr A 13 B A 23 B 16-5 Binomial Probability and the Normal Curve (pages 706–708) 1 91 d. 216 r55 13. 5C5 (.92) (.08) 5 .6591 14. 4C4 (.65)4 (.35)0 5 .1785 0 2 c. 27 c. 0.0322 f. 0.4019 20C2 2 b. 215 216 r 202r 8. a 20Cr A 23 B A 13 B 12. 5 4. a. 25 27 20 < .3073 18 31 d. 32 c. .3125 f. .03125 C (.2)1(.8)2 .3840 3 1 C (.95)3(.05)1 .1715 4 3 a. .2 b. 3C1 (.2)1(.8)2 5 .384 2 3 C (.04) (.96) 5 .0142 5 2 2 13 c. 16 r 102r 10 7. a 10Cr A 12 B A 12 B 5 a 10Cr A 12 B 8. 9. 10. 11. A 121 B A 11 12 B b. 12 r 152r 15 6. a 15Cr A 12 B A 12 B 5 a 15Cr A 12 B without replacement and so the choices are not independent. 2. No, nCr 5 r!(n n!2 r)! 2 n! r! . Developing Skills 3. a. .15625 b. .3125 d. .15625 e. .03125 g. Two or three heads 4. a. 0.4019 b. 0.1608 d. 0.0032 e. 0.0001 g. One or zero sixes 5. a. 0.2420 b. 0.0302 d. 0.00004 e. 0.7260 6. a. 0.0584 b. 0.1877 Applying Skills 3 3. a. 16 10 Successes 361 14580AK05.pgs 3/26/09 12:07 PM 11. Page 362 1 1 1 1 1 1 1 1 5 6 1 7 8 9 6 15 29 1 4 10 20 35 70 126 1 5 15 35 56 85 1 3 10 21 37 2 3 4 1 26. 24 2 4i 27. 9C3 x6(2y)3 5 284x6y3 28. 10C6 a4(3)6 5 153,090a4 29. 12C6 (2x)6(21)6 5 59,136x6 1 1 6 21 Exploration (pages 714–715) 1 7 a. This calculation assumes that 7 games will be played and is the probability that the American League team will win 4 games. The World Series is won by the first team to win 4 games, and so as few as 4 games may be played. b. 4C4 p4 5 p4 c. 4C1 p4(1 2 p) 5 4p4(1 2 p) d. 5C2 p4(1 2 p)2 5 10p4(1 2 p)2 e. 6C3 p4(1 2 p)3 5 20p4(1 2 p)3 f. p4 1 4p4(1 2 p) 1 10p4(1 2 p)2 1 20p4(1 2 p)3 1 56 29 8 1 126 85 37 9 1 12. 15C2x13y2 5 105x13y2 13. 10C6x4 y6 5 210x4y6 14. 6C3(2x)3(y)3 5 160x3y3 15. 9C4(x)5(2y)4 5 126x5y4 16. 7C5(3a)2(2b)5 5 6,048a2b5 17. 8C3(y) 5 A 2y1 B 5 256y2 3 18. 6th term, 7th term 5 1,792a2b6 19. 4th term, 5th term 5 16d4 20. 5th term, 6th term 5 2231x6y5 Applying Skills 21. 27x3 2 27x2 1 9x 2 1 12 Cumulative Review (pages 715–717) Part I 1. 4 4. 1 7. 4 10. 2 12 22. 100 a 12Ci(1) 122i (.01) i 5 100 a 12Ci(.01) i i50 i50 5 11. log2 14 5 log8 x 22 5 log8 x i50 Review Exercises (pages 713–714) 1. 4. 7. 10. 13. 16. 18. 720 2. 20,160 5. 1 8. 132 11. 40,320 14. 120 17. a. .2036 b. c. .7738 d. 33 19. a. 2,048 < .0161 120 28 210 792 19,958,400 56 .0011 .1887 3. 6. 9. 12. 15. 1 x 5 822 5 64 1 1 50 40 360 12. Answer: x 5 1 !3x 1 1 5 3x 1 1 5 05 05 24. 25. Check x 5 1 79 b. 4,096 < .0193 ? !3(1) 1 1 5 2(1) c. 4,096 < .9968 .868 a. .021 b. .023 c. .683 x4 1 4x3y 1 6x2y2 1 4xy3 1 y4 128a7 1 448a6 1 672a5 1 560a4 1 280a3 1 84a2 1 14a 1 1 729 2 1,458x 1 1,215x2 2 540x3 1 135x4 2 18x5 1 x6 b8 2 8b6 1 28b4 2 56b2 1 70 2 562 1 284 2 86 1 18 b b 2x 4x2 4x2 2 3x 2 1 (x 2 1)(4x 1 1) x 5 1, 214 4,083 20. 21. 22. 23. 3. 4 6. 1 9. 3 Part II 5 23. 75,000 a 5Ci(1) 52i (2.20) i 5 75,000 a 5Ci(2.20) i i50 2. 3 5. 4 8. 1 ? !4 5 2 2 5 2✔ b b 362 Check x 5 214 ? 1 1 #3 A 24 B 1 1 5 2 A 24 B #4 5 22 1 ? 1 2 1 2 212 ✘ 14580AK05.pgs 3/26/09 12:07 PM Page 363 Part IV 15. a. Volunteers Part III 13. a–b. y 1 O 1 x 120 100 80 60 40 20 0 0 2 4 6 8 10 Week x 5 log1.6 x f21 (x) 5 lnln(1.6) 5 14. Answer: 22 , w , 4 b. y 5 31.1327 1 31.1523 ln x 16. l 5 2w 2 3 c. 103 2 cos2 u 1 2 sin u 2 1 5 0 2(1 2 sin2 u) 1 2 sin u 2 1 5 0 A 5 lw 5 2w2 2 3w , 20 22 sin2 u 1 2 sin u 1 1 5 0 2w2 2 3w 2 20 , 0 !4 1 8 sin u 5 22 6 24 (2w 1 5)(w 2 4) , 0 sin u 5 12 6 !3 2 The solutions to the corresponding equality are 252 and 4. u 5 arcsin Q 1 22 !3 R The original inequality is true in the interval 252 , w , 4. u 5 2018, 3398 363