Download Equation Chapter 1 Section 1MATH 209

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MATH 209 INDIVIDUAL TEXTBOOK
SOLUTIONS WEEK 1
Note: Application problems are marked in green.
Exponents & Polynomials (Chapter 4 of Dugo 2nd Ed)
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Section 4.1: Exercises 50, 66, 72,90, 92, and 94
Section 4.2: Exercises 26, 40, 58, 60, 62, 68, 72, 84, 110, 112, and 114
Section 4.3: Exercises 32, 48, 64, 94, 96, 98, and 100
Section 4.4: Exercises 16, 34, 68, 78, 80, and 82
Section 4.5: Exercises 28, 40, 52,98, 100, and 102
Section 4.6: Exercises 48, 74, 78, 94, 96,and 98
Section 4.7: Exercises 12, 24, 26, 66, 88, and 90
Section4.1,#50
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Section4.1,#66
Section 4.1 #90 Alberto invested $80,000 in his brother’s restaurant. His brother
did well and paid him back after 5 years with 10% interest compounded annually.
What was the amount that Alberto received?
Using the “Amount Formula”on page 261
A=P(1+r)n
where P= $80,000
r = 0.10
and n = 5
Find the value of A
A=80,000(1+0.10) 5 , which is the same as
A=80,000(1.1∙1.1∙1.1∙1.1∙1.1) = 80,000(1.61) =128,841
Alberto received $128,841 (rounded off to the nearest $)
Section 4.1 #92 - Helene put her $30,000 inheritance into a savings account at her
bank and earned 2.2% compounded annually for 10 years. How much did she
have after the tenth year?
A=P(1+r)n
where P= $30,000
r = 0.022
And n = 10
Find the value of A
A=30,000(1+0.022) 10 = 30,000(1.243) = 37293
Helene earned $37, 293 (rounded off to the nearest $)
, Section 4.1 #94 - Ronnie invested P dollars in a 2-year CD with an annual return
of r. After the CD rolled over three times, its value was P ((1+ r) 2)3 dollars. Which
rule of exponents can be used to simplify the expression?
Power of a power Rule
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Simplify it
P((1+ r)2)3 = P(1+r)6
Section 4.2 #26
=
Section 4.2 #40
Section 4.2 #58
Section 4.2 #60 -Mr. Isaacs wants to have $60,000 in 18 years when little Debby
will start college. How much would he have to invest today in high-yield bonds
that pay 9% compounded annually to achieve his goal?
A =$60000
r=9%=0.09
n=18
Mr. Isaacs must invest $12,720 to realize his objective.
Section 4.2 #62 - Oscar has an account that is earmarked for a sailboat. He needs
$200,000 for the boat when he retires in 10 years. If he averages 7% annually on
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this account, how much should he have in the account now so that his goal will
be reached with no additional deposits.
Using the same formula as in #60 above:
Oscar should set aside $101,670
Section 4.2 #68
9.3 x 10-5= 0.000093
Section 4.2 #72
8x106 = 8,000,000
Section 4.2 #84
Section 4.2 #110 - The distance from the earth to the sun is 93 million miles. The
speed of light is 9.83569 x108 feet per second. How long does it take light to
travel from the sun to the earth?
93 x106 milesx5.28x103 ft = 491.04x109ft = 4.9104x1011ft
1 mile
Distance formula is Distance = Speed x Time
D=RT
Then T=D/R
T =4.9104x1011ft /(9.83569x108)ft/sec = 0.4992x1011-8 =0.4992x103
=499.2 seconds
Section 4.2 #112 - If the radius of a very small circle is 2.35 x 10-8 centimeters,
then what is the circle’s area?
A=π r2 = 3.1416(2.35x10-8)2 = 3.1414(5.52x10-16) =17.34x10-16
=1.734x10-15 square centimeters
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Section 4.2 #114 - If the diameter of a circle is 1.3x 10-12 meters, then what is its
radius?
r = d/2
1.3x10-12/2 = 0.65x10-12 = 6.5x10-13 meters
Section 4.3 #32
Solve -2x4 - 3x2+ 5x - 9 for x=2.
-2(2)4 -3(2)2 +5(2) – 9 = -32 -12 +10 – 9 = -43
Section 4.3 #48
Simplify
(w2 - 2w+ 1)+ (2w- 5 +w2) = w2 - 2w+ 1+ 2w- 5 +w2 =2w2 -4
Section 4.3 #64
(4 - 5y+ y3) - (2- 3y + y2) = 4 - 5y+ y3 – 2+ 3y - y2 = y3 –y2 -2y + 2
Section 4.3 #94 - The width of a rectangular playground is 2x -5 feet, and the
length is 3x + 9 feet. Write a polynomial P(x) that represents the perimeter and
then evaluate this perimeter polynomial if x is 4 feet.
P = 2(2x-5) +2(3x+9) = 4x-10 + 6x + 18
P = 10x +8
When x=4ft
P=10(4) + 8 = 48 ft.
Section 4.3 #96 - Jessica traveled 2x + 50 miles in the morning and 3x - 10 miles in
the afternoon. Write a polynomial T(x) that represents the total distance that she
traveled. Find the T(20).
T(x) = 2x+50 + 3x – 10
T(x) = 5x +40
T (20) – 5(20) +40 = 140 miles
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Section 4.3 #98 - A red ball and a green ball are simultaneously tossed into the air.
The red ball is given an initial velocity of 96 feet per second, and its height t
seconds after it is tossed is -16t2 + 96t feet. The green ball is given an initial
velocity of 80 feet per second, and its height t seconds after it is tossed is
-16t2 + 80t feet.
a) Find a polynomial D (t) that represents the difference in the heights of the two
balls.
D (t) = is -16t2 + 96t – (-16t2 + 80t) = -16t2 + 96t +16t2 - 80t = 16t
b) How much higher is the red ball 2 seconds after the balls are tossed?
D (2) = 32 ft
c) In reality, when does the difference in the heights stop increasing?
When the last ball (green) hits the ground
Section 4.3 #100 - Deborah figured that the amount of acid in one bottle of solution
is 0.12x milliliters and the amount of acid in another bottle of solution is 0.22(75x) milliliters. Find a polynomial T(x) that represents the total amount of acid?
What is the total amount of acid if x = 50?
T(x) = 0.12x +0.22(75-x) = 0.12x + 16.5 -0.22x
T(x) = 16.5 – 0.10x milliliters
T(50) = 16.5 – 0.10(50) = 16.5 -5 = 11.5 milliliters
Section 4.4 #16
-12sq * 3s = -36s2q
Section 4.4 #34
(3c2d-d3+1)8cd2 = 24c3d3 – 8cd5 +8cd2
Section 4.4 #68
(5x-6)(5x-6) = 25x2 -30x – 30x+36 = 25x2 – 60x +36
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Section 4.4 #78 - The length of a rectangular swimming pool is 2x - 1 meters, and
the width is x + 2 meters. Write a polynomial A(x) that represents the area.
Find A (5).
A(x) = (2x-1)(x+2) = 2x2 –x +4x -2
A(x) = 2x2 +3x -2
A (5) = 2(25) +3(5) -2 = 63 square meters
Section 4.4 #80 - The length, width, and height of a box are x, 2x, and 3x - 5
inches, respectively. Write a polynomial V(x) that represents its volume.
Find V(3).
V(x) = x(2x)(3x-5) = 6x3 -10x2 cubic inches
V (3) = 6(27) -10(9) = 162 – 90 = 72 in3
Section 4.4 #82 - If two numbers have a sum of 9, then what polynomial
represents their product?
Product = x(9-x) = 9x – x2
Section4.5 #28
(11x+3y)(x+4y) = 11x2 +44x + 3xy + 12y2 = 11x2 +47xy + 12y2
Section4.5 #40
(5y3w2+z)(2y3w2+3z) =10y6w4 15y3w2z + 2y3w2z + 3z2 = 10y6w4 + 17y3w2z + 3z2
Section4.5 #52
(3h-5)(3h+5) = 9h2 - 25
Section4.5 #98 - Find a trinomial A(x) that represents the area of a parallelogram
whose base is 3x+ 2 meters and whose height is 2x + 3 meters.
Find A(3).
A(x) = (3x+2)(2x+3) = 6x2 + 13x + 6 square meters
A(3) = 6(9) +13(3) +6 = 99m2
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Section4.5 #100 - A square has sides of length 3x + 1 meters. Find a polynomial
A(x) that represents the area of the square.
Find A (1).
A(x) = (3x+1)2 = 9x2 +6x + 1
A(1) = 9(1) +6(1) + 1 = 16 m2
Section4.5 #102 - Find the area of each of the four regions shown in
the figure in the text. What is the total area of the four regions?
A = h2 +4h + 12 + 3h = h2 +7h + 12 = (h+4) (h+3)
What does this exercise illustrate?
The factoring process
Section4.6 #48
(3y2 + 1)(3y2 - 1) = 9y2 -1
Section4.6 #74
(3z4 – 8)2 = 9z8 - -48z4 + 64
Section4.6 #78
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Section4.6 #94 - A promoter is planning a circular race track with an inside radius
of r feet and a width of w feet. The cost in dollars for paving the track is given by
the formula, C = 1.2π [(r + w) 2 - r 2].
Use a special product rule to simplify this formula. What is the cost of paving the
track if the inside radius is 1000 feet and the width of the track is 40 feet?
C = 1.2 [r2 +2rw + w2 – r2] = 1.2π (2rw +w2)
C = 1.2 (3.1416) (2(1000) (40) +1600) = 3.77(80000 +1600)
= 3.77(81600) = $307,625
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Section4.6 #96 - P dollars is invested at annual interest rate r for 1 year. If the
interest is compounded semiannually, then the polynomial P(1 +r/2)2 represents
the value of the investment after 1 year. Rewrite this expression without
parentheses. Evaluate the polynomial if P =$200 and r =10%.
P (1+r/2)2 = P(1+r+r2/4) = P +Pr+ Pr2/4
For P = $200 and r = 0.10
200 + (200)(0.10) + 50(0.01) = 200 + 20 + 0.5
$220.50
Section4.6 #98 - How much more would the investment in Exercise 97 be worth in
10 years if the client invests in large company stocks rather than U.S. treasury
bills?
Large company stocks yield
10,000(1.167)10 = $46, 849.89
US Treasury Bills yield
10,000(1.073)10 = $20, 230.06
Large company stocks would yield $26,619.83 more than U.S. Treasury bills.
Section4.7 #12
Section4.7 #24
Section4.7 #26
Section4.7 #66
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Section4.7 #88 - The perimeter of a rectangular backyard is 6x + 6 yards. If the
width is x yards, find a binomial that represents the length.
P = 2L + 2W
6x + 6 = 2L +2x
2L = 6x +6 -2x = 4x+6
L = 2x+3
Section4.7 #90 Divide a3-b3 by a - b and a4 - b4 by a- b. What is the quotient when
a8 - b8 is divided by a - b?
(a3-b3)/ (a-b) = a2 +ab+b2
(a4-b4)/ (a-b) = a3+a2b +a2b+ b3
(a8 –b8)/ (a-b) = a7 +a6b +a5b2+a4b3 + a3b4 + a2b5 + ab6 + b7
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