Download 14.4 Exercises Answer Key

Chapter 14
10.
− 2
− 9 = −3
−
11.
− 4 = −2
8. yes; Using a calculator,
− 1 = −1
2 is to the right of −2. So, −
A
=
3
64
=
3
0
2 is greater.
rational number.
16 = 4
25 = 5
36 = 6
1
1
Because 21 is closer to 25 than to 16, 21 is closer to
3
3
5 than to 4. So, the radius is about 5 inches.
h = 1.17 10 ≈ 3.70
Because 3.70 is less than 4, you cannot see farther than
4 nautical miles.
Vocabulary and Concept Check
ratio of two integers. An irrational number cannot be
written as the ratio of two integers.
2. Because 32 is between the perfect squares 25 and 36,
32 ≈ 6.
7 are real
8 does not belong because it is an irrational number.
The other three numbers are rational numbers.
Practice and Problem Solving
559
5. yes; Using a calculator,
= 2.236 and
250
5 ≈ 2.236067977. So, the rational number is a
reasonable approximation of the square root.
6. no; Using a calculator,
3021
= 12.084 and
250
11 ≈ 3.31662479. So, the rational number is not a
reasonable approximation of the square root.
7. no; Using a calculator,
12. Because −
81 = − 9, the number −
81 is an integer
and a rational number.
52
52
is a natural number,
= 4, the number
13
13
a whole number, an integer, and a rational number.
15. The number − 49 is not a perfect cube. So,
678
= 2.712 and
250
28 ≈ 5.291502622. So, the rational number is not a
reasonable approximation of the square root.
3
− 49 is
irrational.
16. The number 15 is not a perfect square. So,
3. Real numbers are the set of rational and irrational
4.
π
≈ 0.523 neither terminates nor repeats.
6
So, it is irrational.
11. The number
14. Because
1. A rational number is a number that can be written as the
1
numbers. Sample answer: − 2, , and
8
numbers.
3
13. The number −1.125 terminates. So, it is rational.
14.4 Exercises (pp. 651–653)
but is closer to 36,
343 = 7, the number 3 343 is a natural
number, a whole number, an integer, and a rational
number.
10. Because
1
12. 1.17
45 ≈ 6.708203932. So, the rational number is a
reasonable approximation of the square root.
9. The number 0 is a whole number, an integer, and a
1
21
3
213
9=3
1677
= 6.708 and
250
15 is
irrational.
17. Because
So,
144 = 12, the number 144 is a perfect square.
144 is rational.
2
2
2
18. no; a + b = c
42 + 62 = c 2
16 + 36 = c 2
52 = c 2
52 =
c2
52 = c
The length of the hypotenuse is 52 inches. Because
52 is not a perfect square, the length of the hypotenuse is
not a rational number.
19. a. If the last digit of your phone number is 0, it is a
whole number. Otherwise it is a natural number.
b. Because a prime number is divisible only by 1 and
itself, it is not a perfect square. So, the square root
of a prime number is irrational.
circumference
πd
c.
=
= π
diameter
d
Because π neither terminates nor repeats, the ratio
of the circumference of a circle to its diameter is
irrational.
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Chapter 14
20. a. Make a table of numbers whose squares are close to
the radicand, 46.
22. a. Make a table of numbers whose squares are close to
the radicand, 61.
Number
5
6
7
8
Number
6
7
8
9
Square of Number
25
36
49
64
Square of Number
36
49
64
81
The table shows that 46 is not a perfect square. It is
between the perfect squares 25 and 49. Because 46 is
closer to 49 than to 36, 46 is closer to 7 than to 6.
25
36
5
6
49
61
64
46 7
36 = 6
8
46 ≈ 7.
So,
So, −
b. Make a table of numbers between 6 and 7 whose
Number
Square of Number
64 = 8
49 = 7
81 = 9
61 ≈ −8.
b. Make a table of numbers between 7 and 8 whose
squares are close to 46.
squares are close to 61.
6.5
6.6
6.7
6.8
42.25
43.56
44.89
46.24
46 is
Because 46 is closer to 46.24 than to 44.89,
closer to 6.8 than to 6.7.
42.25
43.56
44.89
6.5
6.6
6.7
So,
The table shows that 61 is not a perfect square. It is
between the perfect squares 49 and 64. Because 61 is
closer to 64 than to 49, 61 is closer to 8 than to 7.
46.24
46
6.8
46 ≈ 6.8.
21. a. Make a table of numbers whose squares are close to
7.7
7.8
7.9
8.0
59.29
60.84
62.41
64
Number
Square of Number
61 is
Because 61 is closer to 60.84 than to 62.41,
closer to 7.8 than to 7.9.
59.29
60.84
7.7
7.8
61
62.41
64
7.9
8.0
So, − 61 ≈ −7.8.
23. a. Make a table of numbers whose squares are close to
the radicand, 685.
Number
25
26
27
28
Square of Number
625
676
729
784
The table shows that 685 is not a perfect square. It is
between the perfect squares 676 and 729. Because
685 is closer to 676 than to 729, 685 is closer to
26 than to 27.
685
the radicand, 105.
Number
9
10
11
12
Square of Number
81
100
121
144
The table shows that 105 is not a perfect square. It is
between the perfect squares 9 and 10. Because 105 is
closer to 100 than to 121, 105 is closer to 10 than
to 11.
105
625 = 25
So,
676 = 26
729 = 27
784 = 28
81= 9
685 ≈ 26.
b. Make a table of numbers between 26 and 27 whose
100 = 10
121 = 11
144 = 12
So, − 105 ≈ −10.
squares are close to 685.
26.0
Number
26.1
26.2
26.3
Square of Number 676 681.21 686.44 691.69
Because 685 is closer to 686.44 than to 681.21,
685 is closer to 26.2 than to 26.1.
676
681.21
26
26.1
So,
686.44
685 26.2
691.69
26.3
685 ≈ 26.2.
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Chapter 14
b. Make a table of numbers between 10 and 11 whose
25. a. Make a table of numbers whose squares are close to
squares are close to 105.
the radicand,
10.1
Number
10.2
10.3
10.4
Square of Number 102.01 104.04 106.09 108.16
104.04
10.1
10.2
11
12
13
14
Square of Number
121
144
169
196
335
is not a perfect square. It is
2
between the perfect squares 12 and 13. Because
335
335
is
= 167.5 is closer to 169 than to 144,
2
2
closer to 13 than to 12.
The table shows that
106.09
108.16
10.3
10.4
105
Number
105
Because 105 is closer to 104.04 than to 106.09,
is closer to 10.2 than to 10.3.
102.01
335
= 167.5.
2
So, − 105 ≈ −10.2.
335
2
24. a. Make a table of numbers whose squares are close to
the radicand,
27
.
4
121 = 11
1
2
3
4
Square of Number
1
4
9
16
b. Make a table of numbers between 12.5 and 13.5
whose squares are close to
27
is not a perfect square. It is
The table shows that
4
between the perfect squares 2 and 3. Because
27
= 6.75 is closer to 9 than to 4,
4
3 than to 2.
27
is closer to
4
So,
b. Make a table of numbers between 2 and 3 whose
27
.
4
2.4
2.5
2.6
2.7
Square of Number
5.76
6.25
6.76
7.29
166.41
12.8
12.9
26.
27
is
4
27
is closer to 6.76 than to 2.5,
4
closer to 2.6 than to 2.5.
So,
5.76
6.25
2.4
2.5
27
4
6.76
7.29
2.6
2.7
25 = 5
13.1
36 = 6
49 = 7
64 = 8
81 = 9
100 = 10
20. So, 10 is greater.
15
− 4 = −2
4=2
0
15 is to the right of −3.5 because
−3.5 is negative. So,
28.
16 = 4
15 is positive and
15 is greater.
3
10 4 = 10.75
100 = 10
121 = 11
133
144 = 12
3
133 is to the right of 10 . So,
4
Worked-Out Solutions
13.0
−3.5
− 16 = −4
27
≈ 2.6.
4
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171.61
335
≈ −12.9.
2
10 is to the right of
27.
335
2
169
20
16 = 4
Because
13.1
335
= 167.5 is closer to 166.41 than to 169,
2
163.84
So, −
Number
13.0
335
is closer to 12.9 than to 13.0.
2
16 = 4
27
≈ 3.
4
squares are close to
12.9
Square of Number 163.84 166.41 169 171.61
Because
9=3
4=2
335
.
2
12.8
Number
27
4
1=1
196 = 14
169 = 13
335
≈ −13.
2
So, −
Number
144 = 12
169 = 13
133 is greater.
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Chapter 14
29.
16
=4
81
9
37. a 2 + b 2 = c 2
2
3
0
1=1
1
1
=2
4
2
is to the right of
3
30.
3
1
8.5 ≈ c
0
0.25. So, −0.25 is greater.
38. a 2 + b 2 = c 2
4 2 + 82 = c 2
− 182
− 196 = −14
32. false; To the nearest tenth,
16 + 64 = c 2
− 169 = −13
182 is to the right of − 192. So, −
80 = c 2
− 144 = −12
80 =
182 is greater.
35.
So, the approximate length of the diagonal is
8.9 centimeters.
39.
10 is greater than 3.16.
a 2 + b2 = c2
102 + 182 = c 2
A = s2
100 + 324 = c 2
66 = s 2
424 = c 2
66 =
424 =
s2
c2
20.6 ≈ c
66 = s
So, the approximate length of the diagonal is 20.6 inches.
66
64 = 8
67.24 = 8.2
65.61 = 8.1
68.89 = 8.3
Because 66 is closer to 65.61 than to 67.24, 66 is
closer to 8.1 than to 8.2. So, one of the sides of the court
is about 8.1 feet.
36.
c2
8.9 ≈ c
10 ≈ 3.2.
33. true
34. false;
c2
So, the approximate length of the diagonal is 8.5 feet.
1
− 192
−
72 =
−4
−0.25 is to the right of −
− 225 = −15
72 = c 2
−0.25
−2
31.
36 + 36 = c 2
16
2
. So, is greater.
81
3
− 0.25 = −0.5
−4
62 + 62 = c 2
9
1
= 12
4
71 to the nearest hundredth, create a table
of numbers between 8.4 and 8.5 whose squares are close
to 71, and then determine which square is closest to 71.
40. To estimate
A = s2
14 = s 2
14 =
s2
14 = s
14
12.96 = 3.6
13.69 = 3.7
14.44 = 3.8
15.21 = 3.9
Because 14 is closer to 13.69 than to 14.44,
closer to 3.7 than to 3.8. So,
14 is
14 ≈ 3.7 and s ≈ 3.7.
The side length of a square on the checkerboard is about
3.7 centimeters. The sides of the checkerboard contain
8 squares each. So, the length of a side of the
checkerboard is x ≈ 8(3.7) = 29.6 centimeters.
P = 4 x ≈ 4( 29.6) = 118.4
The perimeter of the checkerboard is about
118.4 centimeters.
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Chapter 14
41. To estimate a cube root to the nearest tenth, create a table
of integers whose cubes are close to the radicand.
Determine which two integers the cube root is between.
Then create another table of numbers between those two
integers whose cubes are close to the radicand. Determine
which cube is closest to the radicand.
2
1 1
1
= , is a perfect square. So,
4 4
 2
is a rational number.
48. yes; Because  
3
is not a perfect square,
16
rational number.
no; Because
Make a table of numbers whose cubes are close to the
radicand, 14.
Number
1
2
3
4
Cube of Number
1
8
27
64
49. t =
The table shows that 14 is not a perfect cube. It is
between the perfect cubes 2 and 3.
Make a table of numbers between 2 and 3 whose cubes
are close to 14.
2.3
Number
2.4
2.5
closer to 2.4 than to 2.5. So,
42. x = 1.23
3
be written as the ratio of two integers. So, the product
is rational.
b. sometimes; The product of a nonzero rational number
and an irrational number cannot be written as the ratio
of two integers. However, the product of 0 and an
irrational number is 0, which is rational.
14 is
14 ≈ 2.4.
Sample answer: 3 • π = 3π is irrational, but
0 • π = 0 is rational.
The maximum distance is about 182.4 nautical miles.
c. sometimes; The product of two irrational numbers can
be written or cannot be written as the ratio of two
integers.
81 = 9 and 100 = 10, a
and b are any numbers between 81 and 100, and b > a.
So, one answer is a = 82 and b = 97.
43. Sample answer: Because
Sample answer: π •
π
0.39
0.25 = 0.5
0.36 = 0.6
3
0.49 = 0.7
45.
0.39 is
1 =1
0.39 ≈ 0.6.
to 1.1 than to 1. So,
46.
51.
47. s = 3
c2
40 = c
The length of the hypotenuse is 40 meters.
1.69 = 1.3
1.96 = 1.4
1.52 is
1.52 ≈ 1.2.
6 r = 3 6(16.764) = 3 100.584 ≈ 30.1
The speed of a car going around the loop is about
30.1 meters per second.
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is irrational.
a 2 + b2 = c2
1600 =
1.19 is closer
Because 1.52 is closer to 1.44 than to 1.69,
closer to 1.2 than to 1.3. So,
6
1600 = c 2
1.44 = 1.2
1.19 ≈ 1.1.
1.44 = 1.2
π2
242 + 322 = c 2
1.52
1.21 = 1.1
=
576 + 1024 = c 2
1.21 = 1.1
Because 1.19 is closer to 1.21 than to 1,
2
= 1 is rational.
Fair Game Review
1.19
0.81 = 0.9
•
1
π
π
0.64 = 0.8
Because 0.39 is closer to 0.36 than to 0.49,
closer to 0.6 than to 0.7. So,
3 1
3
• = −
4 5
20
Sample answer: −
h = 1.23 22,000 ≈ 182.4
44.
14
≈ 1.7
4.9
50. a. always; The product of two rational numbers can
Cube of Number 12.167 13.824 15.625 17.576
3
3
is not a
16
It takes the balloon about 1.7 seconds to fall to the
ground.
2.6
Because 14 is closer to 13.824 than to 15.625,
d
=
4.9
1
4
52.
a 2 + b2 = c2
102 + b 2 = 262
100 + b 2 = 676
b 2 = 576
b2 =
576
b = 24
The length of the leg is 24 inches.
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Chapter 14
53.
a 2 + b2 = c2
2
2
a + 12 = 15
4. Let x = 5.8.
2
x = 5.8
a 2 + 144 = 225
10 • x = 10 • 5.8
a 2 = 81
a
2
=
10 x = 58.8
− (x = 5.8 )
81
9 x = 53
a = 9
The length of the leg is 9 centimeters.
54. D;
side length of red triangle
4
2
=
=
side length of blue triangle
10
5
x =
53
8
= 5
9
9
8
So, 5.8 = 5 .
9
The ratio is 2 : 5.
14.4 Extension (pp. 654–655)
1. Let x = 0.1.
x = 0.1
5. Because the solution does not change when
adding/subtracting two equivalent equations; Multiply by
10 so that when you subtract the original equation, the
repeating part is removed.
6. Write the digit that repeats in the numerator and use 9 in
10 • x = 10 • 0.1
the denominator.
10 x = 1.1
7. Let x = − 0.43.
− (x = 0.1)
9x = 1
x = − 0.43
1
x =
9
10 • x = 10 • − 0.43
So, 0.1 =
(
10 x =
1
.
9
)
− 4.3
− (x = − 0.43)
9 x = − 3.9
2. Let x = − 0.5.
x = −
x = − 0.5
(
)
10 • x = 10 • − 0.5
So, 0.43 = −
10 x = − 5.5
3.9
9
3.9
39
13
= −
= − .
9
90
30
8. Let x = 2.06.
− (x = − 0.5 )
9x =
−5
x = 2.06
x =
5
−
9
10 • x = 10 • 2.06
10 x = 20.6
− (x = 2.06 )
5
So, − 0.5 = − .
9
9 x = 18.6
3. Let x = −1.2.
x =
x = −1.2
(
10 • x = 10 • −1.2
)
So, 2.06 =
18.6
9
18.6
186
6
1
=
= 2
= 2 .
9
90
90
15
10 x = −12.2
− (x =
9x =
−1.2 )
−11
x = −
11
2
or −1 .
9
9
2
So, −1.2 = −1 .
9
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