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Appendix C Answers to Selected Exercises

1 Linear Models
1.1 Creating a Linear Model
1.1.3 Problem Set 1.1

Applications

1.1.3.13.
Answer.
  1. Altitude (1000 ft) 0 1 2 3 4 5
    Boiling point (F) 212 210 208 206 204 202
  2. boiling point
  3. 4F
  4. Over 4000 feet

1.2 Graphs and Equations
1.2.6 Problem Set 1.2

1.3 Intercepts
1.3.6 Problem Set 1.3

Skills Practice

1.3.6.13.
Answer.
2x+3y=2400

Applications

1.3.6.23.
Answer.
  1.  t  0 5 10 15 20
     h  400 300 200 100 0
  2. h=44+20t
  3. grid
  4. (0,400): The diver starts at a depth of 400 feet. (20,0): The diver surfaces after 20 minutes.
1.3.6.25.
Answer.
  1. $2.40x, $3.20y
  2. 2.40x+3.20y=19,200
  3. line
  4. The y-intercept, 6000 gallons, is the amount of premium that the gas station owner can buy if he buys no regular. The x-intercept, 8000 gallons, is the amount of regular he can buy if he buys no premium.

1.4 Slope
1.4.5 Problem Set 1.4

Applications

1.4.5.25.
Answer.
  1. Yes, the slope is 0.12
  2. 0.12 cm/kg: The spring stretches an extra 0.12 cm for each additional 1 kg mass.
1.4.5.27.
Answer.
  1. The distances to the stations are known.
  2. 5.7 km/sec

1.5 Equations of Lines
1.5.6 Problem Set 1.5

Warm Up

1.5.6.1.
Answer.
x 1  0   2   3   4 
y 10 8 4 2 0
line
  1. decreases by 2 units
  2. The constant term is the y-intercept and the coefficient of x is the slope.

Applications

1.5.6.19.
Answer.
  1. M=7000400w
  2. The slope tells us that Tammy’s bank account is diminishing at a rate of $400 per week, the vertical intercept that she had $7000 (when she lost all sources of income).
1.5.6.21.
Answer.
m=0.0018 degree/foot, so the boiling point drops with altitude at a rate of 0.0018 degree per foot. b=212, so the boiling point is 212 at sea level (where the elevation h=0).
1.5.6.25.
Answer.
  1. C 15 5
    F 59 23
  2. F=32+95C
  3. m=95, so an increase of 1C is equivalent to an increase of 95F.
1.5.6.27.
Answer.
  1. 55F
  2. 9840 ft
  3. temperature vs altitude
  4. m=3820. The temperature decreases 3 degrees for each increase in altitude of 820 feet.
  5. (19,13313,0). At an altitude of 19,13313 feet, the temperature is 0F. (0,70). At an altitude of 0 feet, the temperature is 70F.

1.6 Chapter Summary and Review
1.6.3 Chapter 1 Review Problems

1.6.3.2.

Answer.
  1. t  5   10   15   20   25 
    R 1960 1820 1680 1540 1400
  2. R=210028t
  3. (75,0), (0,2100)
    grid
  4. t-intercept: The oil reserves will be gone in 2080; R-intercept: There were 2100 billion barrels of oil reserves in 2005.

1.6.3.3.

Answer.
  1. 60C+100T=1200
  2. (20,0), (0,12)
    grid
  3. 6 days
  4. She can spend 20 days in Atlantic City if she spends no time in Saint-Tropez, or 12 days in Saint-Tropez if she spends no time in Atlantic City.

1.6.3.4.

Answer.

1.6.3.5.

Answer.

1.6.3.6.

Answer.

1.6.3.7.

Answer.

1.6.3.8.

Answer.

1.6.3.9.

Answer.

1.6.3.10.

Answer.
  1. B=8005t
  2. GC graph
  3. m=5 thousand barrels/minute: The amount of oil in the tanker is decreasing by 5000 barrels per minute.

1.6.3.11.

Answer.
  1. F=500+0.10C
  2. GC graph
  3. m=0.10: The fee increases by $0.10 for each dollar increase in the remodeling job.

1.6.3.18.

Answer.
d V
5 4.8
2 3
1 1.2
6 1.8
10 4.2

1.6.3.19.

Answer.
q S
8 8
4 36
3 168
5 200
9 264

1.6.3.21.

Answer.
m=12, b=54

1.6.3.23.

Answer.
m=4, b=3

1.6.3.30.

Answer.
  1. t 0 15
    P 4800 6780
  2. P=4800+132t
  3. m=132 people/year: the population grew at a rate of 132 people per year.

2 Applications of Linear Models
2.1 Linear Regression
2.1.5 Problem Set 2.1

Warm Up

2.1.5.1.
Answer.
a: II; b: III; c: I; d: IV
2.1.5.2.
Answer.
a: III; b: IV; c: II; d: I
2.1.5.3.
Answer.
  1. k 70 50
    p 154 110
  2. p=2.2k
  3. m=2.2 pounds/kg is the conversion factor from kilograms to pounds

Skills Practice

2.1.5.5.
Answer.
The slope is 2.5, which indicates that the snack bar sells 2.5 fewer cups of cocoa for each 1C increase in temperature. The C-intercept of 52 indicates that 52 cups of cocoa would be sold at a temperature of 0C. The T-intercept of 20.8 indicates that no cocoa will be sold at a temperature of 20.8C.
2.1.5.7.
Answer.
2.1.5.9.
Answer.
  1. L=24+(29/7)t, where t is in months
  2. 74 feet
  3. The young whale grows in length about 4.14 foot per month
2.1.5.11.
Answer.
2 min: 21C; 2 hr: 729C; The estimate at 2 minutes is reasonable; the estimate at 2 hours is not reasonable.

Applications

2.1.5.19.
Answer.
  1. scatterplot and regression line
  2. The graph is above.
  3. The slope tells us that the time it takes for a bird to attract a mate decreases by 0.85 days for every additional song it learns.
  4. 44.5 days
  5. The C-intercept tells us that a warbler with a repretoire of 53 songs would acquire a mate immediately. The B-intercept tells us that a warbler with no songs would take 62 days to find a mate. These values make sense in context.

2.2 Linear Systems
2.2.5 Problem Set 2.2

Warm Up

2.2.5.1.
Answer.

Skills Practice

Applications

2.2.5.13.
Answer.
  1. Sporthaus: y=500+10x
    Fitness First: y=50+25x
  2. x Sporthaus Fitness First
    6 560 200
    12 620 350
    18 680 500
    24 740 650
    30 800 800
    36 860 950
    42 920 1100
    48 980 1250
  3. Cost for fitness clubs
  4. 30 months
2.2.5.19.
Answer.
  1. The median age
  2. grid
  3. 0.3 years of age per year since 1990
  4. See (b) above
  5. Slightly less than 14 years since 1990
  6. More than half the women are older than the mean age of women.

2.3 Algebraic Solution of Systems
2.3.5 Problem Set 2.3

Warm Up

Applications

2.3.5.11.
Answer.
  1. S=2.5x;  D=3504.5x
  2. 50 dollars per machine; 125 machines
2.3.5.13.
Answer.
  1. Pounds % silver Amount of silver
    First alloy x 0.45 0.45x
    Second alloy y 0.60 0.60y
    Mixture 40 0.48 0.48(40)
  2. x+y=40
  3. 0.45x+0.60y=19.2
  4. 32 lb
2.3.5.15.
Answer.
  1. Rani’s speed in still water: x
    Speed of the current: y
    Rate Time Distance
    Downstream x+y 45 6000
    Upstream xy 45 4800
  2. 45(x+y)=6000
  3. 45(xy)=4800
  4. Rani’s speed in still water is 120 meters per minute, and the speed of the current is 1313 meters per minute.
2.3.5.19.
Answer.
  1. Sports coupes Wagons Total
    Hours of riveting 3 4 120
    Hours of welding 4 5 155
  2. 3x+4y=120
  3. 4x+5y=155
  4. 20 sports coupes and 15 wagons

2.4 Gaussian Reduction
2.4.6 Problem Set 2.4

Warm Up

2.4.6.3.
Answer.
  1. Principal Interest rate Interest
    Bonds x 0.10 0.10x
    Certificate y 0.08 0.08y
    Total 2000 —— 184
  2. x+y=2000
  3. 0.10x+0.08y=184

Skills Practice

2.4.6.11.
Answer.
(12,23,3)
2.4.6.13.
Answer.
(12,12,13)

Applications

2.4.6.17.
Answer.
x=40 in, y=60 in, z=55 in
2.4.6.19.
Answer.
12 lb Colombian, 14 lb French, 14 Sumatran
2.4.6.21.
Answer.

2.5 Linear Inequalities in Two Variables
2.5.5 Problem Set 2.5

Warm Up

2.5.5.3.
Answer.
  1. The graph of the equation is a line, and the graph of the inequality is a half-plane. The line is the boundary of the half-plane but is not included in the solution to the inequality.
  2. The graph of x+y10,000 includes both the line x+y=10,000 and the half-plane of the corresponding strict inequality.

Skills Practice

2.5.5.5.
Answer.
2.5.5.7.
Answer.
2.5.5.9.
Answer.
2.5.5.11.
Answer.
2.5.5.13.
Answer.
2.5.5.15.
Answer.
2.5.5.17.
Answer.
2.5.5.19.
Answer.
2.5.5.21.
Answer.
2.5.5.23.
Answer.

Applications

2.5.5.25.
Answer.
2.5.5.27.
Answer.

2.6 Chapter Summary and Review
2.6.3 Chapter 2 Review Problems

2.6.3.11.

Answer.
Consistent and independent

2.6.3.12.

Answer.
Inconsistent

2.6.3.14.

Answer.
Consistent and independent

2.6.3.23.

Answer.
$3181.82 at 8%, $1818.18 at 13.5%

2.6.3.25.

Answer.
5 cm, 12 cm, 13 cm

2.6.3.26.

Answer.
20 to Boston, 25 to Chicago, 10 to Los Angeles

2.6.3.27.

Answer.

2.6.3.28.

Answer.

2.6.3.29.

Answer.

2.6.3.30.

Answer.

2.6.3.31.

Answer.

2.6.3.32.

Answer.

2.6.3.33.

Answer.

2.6.3.34.

Answer.

2.6.3.35.

Answer.

2.6.3.36.

Answer.

2.6.3.37.

Answer.

2.6.3.38.

Answer.

2.6.3.39.

Answer.
20p+9g120, 10p+10g120, p0, g0
system of inequalities

2.6.3.40.

Answer.
x+y32, 2x+1.6y56, x0, y0, where x represents ounces of tofu, y the ounces of tempeh
system of inequalities

3 Quadratic Models
3.1 Extraction of Roots
3.1.9 Problem Set 3.1

3.2 Intercepts, Solutions, and Factors
3.2.6 Problem Set 3.2

Skills Practice

3.2.6.25.
Answer.
0.1(x18)(x+15)
3.2.6.27.
Answer.
All three graphs have the same x-intercepts.

Applications

3.2.6.29.
Answer.
c. 306.5 ft at 0.625 sec d. 1.25 sec e. 5 sec
3.2.6.31.
Answer.
b. h2+102=(h+2)2 c. 24 ft
3.2.6.33.
Answer.
  1. l=x4, w=x4, h=2, V=2(x4)2
  2. 0 cubic in, 2 cubic in, 8 cubic in, etc.
  3. x=4
  4. 9 in by 9 in
  5. 2(x4)2=50; x=9
3.2.6.34.
Answer.
a. l=6, w=12x, h=x, V=6x(12x)   e. 6x(12x)=34;  14  ft

3.3 Graphing Parabolas
3.3.6 Problem Set 3.3

Warm Up

3.3.6.4.
Answer.
  1. factoring; 0,10
  2. extraction of roots; ±10
  3. factoring; 12,1
  4. extraction of roots; 52,52

Applications

3.3.6.23.
Answer.
  1. x-intercepts at 0 and 4
  2. x-intercepts at 0 and 4
  3. reflection of (a) about y-axis
  4. reflection of (b) about y-axis

3.4 Completing the Square
3.4.4 Problem Set 3.4

Applications

3.4.4.29.
Answer.
b2±b24c4
3.4.4.33.
Answer.
(15,0),(15,0);(0,255)

3.5 Chapter 3 Summary and Review
3.5.3 Chapter 3 Review Problems

3.5.3.6.

Answer.
11.5, 23.5

3.5.3.11.

Answer.
4x229x24=0

3.5.3.12.

Answer.
9x230x+25=0

3.5.3.13.

Answer.
y=(x3)(x+2.4)

3.5.3.14.

Answer.
y=(x+1.3)(x2)

3.5.3.17.

Answer.
  1. Vertex: (92,814), y-int: (0,0), x-int: (0,0), (9,0)
  2. parabola

3.5.3.31.

Answer.
10810.4 in

3.5.3.34.

Answer.
50 ft by 150 ft

3.5.3.35.

Answer.
A1=x2(12y2+12y2)=x2y2;  A2=(x+y)(xy)=x2y2

3.5.3.36.

Answer.
A1=π(x+y)2πx2πy2=2πxy;  A2=πy(2x)=2πxy

4 Applications of Quadratic Models
4.1 Quadratic Formula
4.1.6 Problem Set 4.1

Skills Practice

4.1.6.5.
Answer.
0.618, -1.618
4.1.6.11.
Answer.
  1. graph
  2. Approximately (1.5,0) and (0.3,0)
  3. 32 and 13. These are the x-intercepts of the graph.
4.1.6.13.
Answer.
  1. (5,0) and (1,0); (3,0); no x-intercepts
  2. 1 and 5; 3; 6±i122. The real-valued solutions are the x-intercepts of the graph. If the solutions are complex, the graph has no x-intercepts.
4.1.6.17.
Answer.
4±1664h32
4.1.6.19.
Answer.
v±v22asa

Applications

4.1.6.23.
Answer.
c. 31.77 mph
4.1.6.27.
Answer.
b. 10h(2h6)=2160   c. 12 ft by 18 ft by 10ft
Note 4.1.19.
According to the latest research and data, there has been an increase in the number of tigers, and now the total number of wild tigers worldwide is 5,574, according to the World Animal Foundation at https://worldanimalfoundation.org/advocate/animal-captivity-statistics/

4.2 The Vertex
4.2.5 Problem Set 4.2

Warm Up

4.2.5.1.
Answer.
  1. t 0 0.25 0.5 0.75 1 0.25 1.5
    h 12 5 0 3 4 3 0
  2. grid
  3. (1,4)
  4. The wrench reaches its greatest height of 4 feet.
  5. The wrench reaches its greatest height 1 second after Francine throws it.

Applications

4.2.5.17.
Answer.
  1. l=40w; A=40ww2
  2. 400 sq yd; 20 yd by 20 yd
4.2.5.25.
Answer.
  1. graph
  2. (4,0), (0,2)
  3.      x<4 4<x<2 x>2
    Y1 + +
    Y2 +
    Y3 + +

4.3 Curve Fitting
4.3.5 Problem Set 4.3

Skills Practice

4.3.5.9.
Answer.
(2,3,4)
4.3.5.13.
Answer.
a=3, b=1, c=2

4.4 Quadratic Inequalities
4.4.6 Problem Set 4.4

Skills Practice

4.4.6.13.
Answer.
x<2 or x>3
4.4.6.19.
Answer.
(,12)(4,)
4.4.6.21.
Answer.
(,2.24)(2.24,)
4.4.6.25.
Answer.
4.8<x<6.2

Applications

4.4.6.27.
Answer.
(,4.2)(2.6,)
4.4.6.29.
Answer.
4\tt<16 sec
4.4.6.33.
Answer.
  1. 60x;  12+12x
  2. y=(60x)(12+12x)
  3. 882 bu; 18 trees
  4. Between 10 and 26, inclusive

4.5 Chapter 4 Summary and Review
4.5.3 Chapter 4 Review Problems

4.5.3.7.

Answer.
one repeated real solution

4.5.3.8.

Answer.
two complex soutions

4.5.3.9.

Answer.
two complex soutions

4.5.3.10.

Answer.
two distinct real soutions

4.5.3.12.

Answer.
  1. 1.5 sec, 11.025 m
  2. 7.056 m
  3. 0.6 sec
  4. 0.5 sec, 2.5 sec
  5. (0,0), (3,0). She leaves the springboard at t=0 seconds and returns to the springboard at t=3 seconds.

4.5.3.13.

Answer.
  1. (12,494), (3,0), (4,0), (0,12)
  2. parabola

4.5.3.14.

Answer.
  1. (14,318), no x-intercepts, (0,4)
  2. parabola

4.5.3.15.

Answer.
  1. (1,5), (1.24,0), (3.24,0), (0,4)
  2. parabola

4.5.3.17.

Answer.
y=0.2(x15)26

4.5.3.23.

Answer.
a=1, b=1, c=6

4.5.3.24.

Answer.
y=12x24x+10

4.5.3.26.

Answer.
  1. y=0.05x20.003x+234.2
  2. (0.03,234.2)  The velocity of the debris at its maximum height of 234.2 feet. The velocity there is actually zero.

4.5.3.27.

Answer.
(,2)(3,)

4.5.3.30.

Answer.
(,13)(2,)

4.5.3.32.

Answer.
(,3)(3,)

5 Functions and Their Graphs
5.1 Functions
5.1.7 Problem Set 5.1

Applications

5.1.7.13.
Answer.
(b), (c), (e), and (f)
5.1.7.25.
Answer.
  1. N(6000)=2000: 2000 cars will be sold at a price of $6000.
  2. decrease
  3. 30,000. At a price of $30,000, they will sell 400 cars.

5.2 Graphs of Functions
5.2.6 Problem Set 5.2

Skills Practice

5.2.6.7.
Answer.
5.2.6.9.
Answer.

Applications

5.2.6.11.
Answer.
  1. 2,0,5
  2. 2
  3. h(2)=0, h(1)=0, h(0)=2
  4. 5
  5. 3
  6. increasing: (4,2) and (0,3); decreasing: (2,0)
5.2.6.13.
Answer.
  1. 0, 12, 0
  2. 56
  3. 56, 16, 76, 116
  4. 1, 1
  5. Max at x=1.5, 0.5, min at x=0.5, 1.5
5.2.6.15.
Answer.
  1. f(1000)=1495: The speed of sound at a depth of 1000 meters is approximately 1495 meters per second.
  2. d=570 or d=1070: The speed of sound is 1500 meters per second at both a depth of 570 meters and a depth of 1070 meters.
  3. The slowest speed occurs at a depth of about 810 meters and the speed is about 1487 meters per second, so f(810)=1487.
  4. f increases from about 1533 to 1541 in the first 110 meters of depth, then drops to about 1487 at 810 meters, then rises again, passing 1553 at a depth of about 1600 meters.
5.2.6.21.
Answer.
  1. 2, 2
  2. 2.8, 0, 2.8
  3. 2.5<q<1.25 and 1.25<q<2.5
  4. 2<q<0 and 2<q
5.2.6.23.
Answer.
  1. g(6)=0, g(6)=0, g(0)=6
  2. none
  3. g(x) is undefined for those x-values

5.3 Some Basic Graphs
5.3.4 Problem Set 5.3

Applications

5.3.4.19.
Answer.
(,0)[0.5,)
5.3.4.21.
Answer.
(b): 2 units down, (c): 1 unit up
5.3.4.23.
Answer.
(b): 1.5 units left, (c): 1 unit right
5.3.4.25.
Answer.
(b): reflected about x-axis, (c): reflected about y-axis
5.3.4.29.
Answer.
  1. horizontal shift of square root y=x
  2. vertical shift of cube root y=x3
  3. vertical shift of absolute value y=|x|
  4. vertical flip of reciprocal y=1x
  5. vertical flip and vertical shift of cube y=x3
  6. vertical flip and vertical shift of inverse-square y=1x2

5.4 Direct Variation
5.4.6 Problem Set 5.4

Skills Practice

5.4.6.3.
Answer.
  1. y=0.3x
  2. x 2 5 8 12 15
    y 0.6 1.5 2.4 3.6 4.5
  3. y doubles also

Applications

5.4.6.9.
Answer.
  1. Price of item 18 28 12
    Tax 1.17 1.82 0.78
    Tax/Price 0.065 0.065 0.065
    Yes; 6.5%
  2. T=0.065p
  3. direct variation
5.4.6.20.
Answer.
  1. Wind resistance quadruples.
  2. It is one-ninth as great.
  3. It is decreased by 19% because it is 81% of the original.

5.5 Inverse Variation
5.5.4 Problem Set 5.5

Warm Up

5.5.4.1.
Answer.
R=13I. Not inverse variation.
5.5.4.3.
Answer.
W=32,000d Not inverse variation.

Applications

5.5.4.13.
Answer.
  1. Width (feet) 2 2.5 3
    Length (feet) 12 9.6 8
    Length×width 24 24 24
    24 square feet
  2. L=24w
  3. inverse variation
5.5.4.22.
Answer.
  1. It is one-fourth the original illumination.
  2. It is one-ninth the illumination.
  3. It is 64% of the illumination.

5.6 Functions as Models
5.6.6 Problem Set 5.6

Applications

5.6.6.21.
Answer.
y=x3 stretched or compressed vertically
cubic
5.6.6.23.
Answer.
y=1x stretched or compressed vertically
reciprocal

Absolute Value

5.6.6.13.
Answer.
x=32 or x=52
5.6.6.17.
Answer.
b=14 or b=10
5.6.6.19.
Answer.
w=132 or w=152
5.6.6.25.
Answer.
92<x<32
5.6.6.27.
Answer.
d2  or  d5
5.6.6.29.
Answer.
All real numbers
5.6.6.33.
Answer.
T3.2  or  T3.3

5.7 Chapter 5 Summary and Review
5.7.3 Chapter 5 Review Problems

5.7.3.1.

Answer.
A function: Each x has exactly one associated y-value.

5.7.3.2.

Answer.
Not a function

5.7.3.3.

Answer.
Not a function: The IQ of 98 has two possible SAT scores.

5.7.3.5.

Answer.
N(10)=7000: Ten days after the new well is opened, the company has pumped a total of 7000 barrels of oil.

5.7.3.6.

Answer.
H(16)=3: At 16 mph, the trip takes 3 hours.

5.7.3.7.

Answer.
F(0)=1,  F(3)=37

5.7.3.8.

Answer.
G(0)=2,  G(20)=123

5.7.3.9.

Answer.
h(8)=6,  h(8)=14

5.7.3.10.

Answer.
m(5)=6,  m(40)=4.8

5.7.3.16.

Answer.
not a function

5.7.3.17.

Answer.
Not a function

5.7.3.19.

Answer.

5.7.3.21.

Answer.

5.7.3.34.

Answer.
14.0625 lumens

5.7.3.37.

Answer.

5.7.3.38.

Answer.

5.7.3.39.

Answer.

5.7.3.41.

Answer.

5.7.3.48.

Answer.
  1. x 0.25 0.50 1.00 1.50 2.00 4.00
    y 4.00 2.00 1.00 0.67 0.50 0.25
  2. y=1x

6 Powers and Roots
6.1 Integer Exponents
6.1.5 Problem Set 6.1

Skills Practice

6.1.5.23.
Answer.
x3+2x1
6.1.5.25.
Answer.
42u1+6u2
6.1.5.27.
Answer.
4x2(x4+4)

Applications

6.1.5.31.
Answer.
  1. x 1 2 4.5 6.2 9.3
    g(x) 1 0.125 0.011 0.0042 0.0012
  2. they decrease
  3. x 1.5 0.6 0.1 0.03 0.002
    f(x) 0.30 4.63 1000 37,037 125×106
  4. they increase
6.1.5.35.
Answer.
  1. d=50f1
  2. The are of the aperture decreases by a factor of 0.5 at each f-stop.

6.2 Roots and Radicals
6.2.9 Problem Set 6.2

Skills Practice

6.2.9.11.
Answer.
14x1/22x1/2+12x
6.2.9.13.
Answer.
x0.5+x0.251

Applications

6.2.9.31.
Answer.
  1. L (feet) 200 400 600 800 1000
    vmax (knots) 18.4 26 31.8 36.8 41.1
  2. maximum velocity
  3. 50.4 knots
  4. maximum and cruise velocities
  5. 31 knots, 62%
6.2.9.36.
Answer.
  1. 6.5×1013 cm; 1.17×1036 cm3
  2. 1.8×1014g/cm3
  3. Element Carbon Potassium Cobalt Technetium Radium
    Mass
    number, A
    14 40 60 99 226
    Radius, r
    (1013 cm)
    3.1 4.4 5.1 6 7.9
  4. cube root

6.3 Rational Exponents
6.3.7 Problem Set 6.3

Skills Practice

6.3.7.25.
Answer.
2x1/2x1/41
6.3.7.29.
Answer.
a2/3+a1/31a1/3
6.3.7.31.
Answer.
x 0 1 2 3 4 5 6
f(x) 0 1 2.5 4.3 6.4 8.5 10.9
g(x) 0 1 2.8 5.2 8 11.2 14.7
power functions

6.4 Working with Radicals
6.4.6 Problem Set 6.4

Warm Up

6.4.6.5.
Answer.
 ab=ab 
 ab=ab 

Skills Practice

6.5 Radical Equations
6.5.6 Problem Set 6.5

6.6 Chapter 6 Summary and Review
6.6.3 Chapter 6 Review Problems

6.6.3.7.

Answer.
  1. 1.018×109 sec, or 0.000 000 001 018 sec
  2. 8 min 20 sec

6.6.3.8.

Answer.
1,200,000,000,000 hr, or 78,904,109,590 yr

6.6.3.10.

Answer.
  1. Planet Density
    Mercury 5426
    Venus 5244
    Earth 5497
    Mars 3909
    Jupiter 1241
    Saturn 620
    Uranus 1238
    Neptune 1615
    Pluto 2355
  2. Mercury, Venus, Earth, and Mars

6.6.3.20.

Answer.
Height: 2.673 in; diameter: 5.346 in

6.6.3.44.

Answer.
r=±3q26q+72

7 Exponential Functions
7.1 Exponential Growth and Decay
7.1.7 Problem Set 7.1

Skills Practice

7.1.7.15.
Answer.
20%, 2%, 7.5%, 100%, 115%
7.1.7.19.
Answer.
The growth factor is 1.2.
x 0 1 2 3 4
Q 20 24 28.8 34.56 41.47
7.1.7.20.
Answer.
The decay factor is 0.8.
w 0 1 2 3 4
N 120 96 76.8 61.44 49.15
7.1.7.21.
Answer.
The decay factor is 0.8.
t 0 1 2 3 4
C 10 8 6.4 5.12 4.10
7.1.7.22.
Answer.
The growth factor is 1.1.
n 0 1 2 3 4
B 200 220 242 266.2 292.82

Applications

7.1.7.23.
Answer.
  1. Years after 2010 0 1 2 3 4
    Windsurfers 1500 1680 1882 2107 2360
  2. S(t)=1200(1.12)t
  3. Windsurfer sales exponential growth
  4. 2644; 5844
7.1.7.24.
Answer.
  1. Years after 1983 0 5 10 15 20
    Value of house 20,000 25,526 32,578 41,579 53,066
  2. V(t)=200,000(1.05)t
  3. house value
  4. $359,171.27; $458,403.66
7.1.7.25.
Answer.
  1. Weeks 0 6 12 18 24
    Bees 2000 5000 12,500 31,250 78,125
  2. P(t)=2000(2.5)t/6
  3. bee population
7.1.7.27.
Answer.
  1. Weeks 0 2 4 6 8
    Mosquitos 250,000 187,500 140,625 105,469 79,102
  2. P(t)=250,000(0.75)t/2
  3. graph
  4. 162,280; 68,504
7.1.7.29.
Answer.
  1. Years 0 3 6 9 12
    Value of boat 15,000 13,500 12,150 10,935 9841.50
  2. V(t)=15,000(0.885)t
  3. motorboat depreciation
  4. $4995.52; $4421.04

7.2 Exponential Functions
7.2.6 Problem Set 7.2

Skills Practice

7.2.6.19.
Answer.
x 3 2 1 0 1 2 3
f(x)=3x 127 19 13 1 3 9 27
g(x)=(13)x 27 9 3 1 13 19 127
two exponentials
7.2.6.21.
Answer.
Because they are defined by equivalent expressions, (b), (c), and (d) have identical graphs.

Applications

7.3 Logarithms
7.3.7 Problem Set 7.3

Skills Practice

7.3.7.21.
Answer.
  1. x=log4(2.5)2.7
  2. x=log2(0.2)2.3

7.4 Properties of Logarithms
7.4.5 Problem Set 7.4

Warm Up

Skills Practice

7.4.5.15.
Answer.
  1. log3(3)+4log3(x)
  2. (1/t)log5(1.1)
  3. Extra \left or missing \right
  4. log2(5)+x

7.5 Exponential Models
7.5.5 Problem Set 7.5

Applications

7.5.5.15.
Answer.
P(t)=2000(2t/5;  14.9%
7.5.5.17.
Answer.
D(t)=D0(0,5t/18;  3.8%
7.5.5.23.
Answer.
  1. D(t)=D0(0.5)t/15
  2. After 89.5 years, or in 2060
7.5.5.29.
Answer.
About 11 years

7.6 Chapter 7 Summary and Review
7.6.3 Chapter 7 Review Problems

7.6.3.5.

Answer.

7.6.3.7.

Answer.

7.6.3.21.

Answer.
log0.3(x+1)=2

7.6.3.27.

Answer.
log(5.1)1.30.5433

7.6.3.29.

Answer.
log(2.9/3)0.70.21

7.6.3.35.

Answer.
logb(x)+13logb(y)2logb(z)

7.6.3.37.

Answer.
43log(x)13log(y)

7.6.3.39.

Answer.
Missing \left or extra \right

7.6.3.43.

Answer.
log(63)log(3)3.77

7.6.3.45.

Answer.
log(50)0.3log(6)7.278

7.6.3.47.

Answer.
log(N/N0)k

8 Polynomial and Rational Functions
8.1 Polynomial Functions
8.1.10 Problem Set 8.1

Skills Practice

8.1.10.5.
Answer.
(b) and (c) are not polynomials; they have variables in a denominator.
8.1.10.17.
Answer.
(a2b)(a2+2ab+4b2)
8.1.10.19.
Answer.
(3a+4b)(9a212ab+16b2)
8.1.10.21.
Answer.
(4t3+w2)(16t64t3w2+w4)

8.2 Algebraic Fractions
8.2.5 Problem Set 8.2

Applications

8.2.5.31.
Answer.
  1. 0p<100
  2. p 0 25 50 75 90 100
    C 0 120 360 1080 3240
  3. curve
    60%
  4. p<80%
  5. p=100: The cost of extracting more ore grows without bound as the amount extracted approaches 100%.
8.2.5.33.
Answer.
  1. 2002x1 square centimeters
  2. 8; If x=13, the area of the cross-section is 8 cm2.

8.3 Operations on Algebraic Fractions
8.3.9 Problem Set 8.3

Skills Practice

8.3.9.11.
Answer.
a(2a1)a+4
8.3.9.13.
Answer.
6x(x2)(x1)2(x28)(x22x+4)
8.3.9.21.
Answer.
(z+2)2z2(2z1)
8.3.9.23.
Answer.
(xy)(4x2+2xy+y2)
8.3.9.33.
Answer.
h2+2h3h+2
8.3.9.35.
Answer.
6xx24x(x2)
8.3.9.39.
Answer.
5k+1k(k3)(k+1)
8.3.9.41.
Answer.
3y3y2(y+1)(2y1)

Applications

8.3.9.51.
Answer.
3q8πRa2q8πR3
8.3.9.53.
Answer.
8t+143t
8.3.9.61.
Answer.
4LR2C4L2C
8.3.9.67.
Answer.
  1. 900400+w hr
  2. 900400w hr
  3. Orville, by 1800w160,000w2 hr

8.4 More Operations on Fractions
8.4.5 Problem Set 8.4

Warm Up

8.4.5.1.
Answer.
2x2+x2x(x1)

Applications

8.4.5.25.
Answer.
PQ and overlineRS: ba; QR and overlineSP: ba

8.5 Equations with Fractions
8.5.8 Problem Set 8.5

Skills Practice

8.5.8.17.
Answer.
We don’t multiply by the LCD in addition problems.

Applications

8.5.8.29.
Answer.
168=72p100p; p=70%
8.5.8.39.
Answer.
Because x=1, dividing by x1 in the fourth step is dividing by 0.

8.6 Chapter 8 Summary and Review
8.6.3 Chapter 8 Review Problems

8.6.3.1.

Answer.
2x311x2+19x10

8.6.3.3.

Answer.
(2x3z)(4x2+6xz+9z2)

8.6.3.23.

Answer.
a22aa2+3a+2

8.6.3.27.

Answer.
9x27+4x21x4

8.6.3.29.

Answer.
x22x21x2

8.6.3.33.

Answer.
3x+12(x3)(x+3)

8.6.3.35.

Answer.
2a2a+1(a3)(a1)

8.6.3.53.

Answer.
(xy)2xy

9 Equations and Graphs
9.1 Properties of Lines
9.1.4 Problem Set 9.1

Warm Up

9.1.4.3.
Answer.
Number Negative
reciprocal
Their
product
23 32 1
52 25 1
6 16 1
4 14 1
1 1 1

9.2 The Distance and Midpoint Formulas
9.2.7 Problem Set 9.2

Warm Up

9.2.7.3.
Answer.
x 4 3 2 1 0 1 2 3 4
y 0 ±7 ±12 ±15 ±4 ±15 ±12 ±7 0
circle of radius 4 centered at origin

Skills Practice

9.2.7.7.
Answer.
distance: 20; midpoint: (0,2)
9.2.7.9.
Answer.
distance: 8; midpoint: (2,1)
9.2.7.11.
Answer.
center: (0,0); radius: 5
9.2.7.13.
Answer.
center: (3,0); radius: 10

Applications

9.2.7.15.
Answer.
15+80+4130.3
9.2.7.23.
Answer.
9.2.7.25.
Answer.
9.2.7.27.
Answer.
9.2.7.29.
Answer.
(x+4)2+y2=20; center: (4,0); radius: 20
9.2.7.31.
Answer.
x2+(y5)2=27; center: (0,5); radius: 27
9.2.7.33.
Answer.
(x32)2+(y+4)2=294
9.2.7.35.
Answer.
(x+3)2+(y+1)2=1

9.3 Conic Sections: Ellipses
9.3.5 Problem Set 9.3

Skills Practice

9.3.5.7.
Answer.
9.3.5.9.
Answer.
9.3.5.11.
Answer.
9.3.5.25.
Answer.
(x1)29+(y6)24=1
9.3.5.27.
Answer.
(x3)225+(y3)216=1

9.4 Conic Sections: Hyperbolas
9.4.6 Problem Set 9.4

Skills Practice

9.4.6.3.
Answer.
9.4.6.5.
Answer.
9.4.6.7.
Answer.
9.4.6.19.
Answer.
16x2y2192x4y+556=0
9.4.6.21.
Answer.
x24y2+10x16y27=0
9.4.6.23.
Answer.
Parabola; vertex (0,2), opens downward, a=12
9.4.6.25.
Answer.
Hyperbola; center (18,1), transverse axis vertical, a2=1565, b2=1516
9.4.6.27.
Answer.
Parabola; vertex (4,2), opens upward, a=14

Applications

9.5 Nonlinear Systems
9.5.4 Problem Set 9.5

Skills Practice

9.5.4.5.
Answer.
(1,12),(4,7)
9.5.4.11.
Answer.
(2,2), (2,2)
9.5.4.13.
Answer.
(2,1), (2,1), (1,2), (1,2)
9.5.4.15.
Answer.
(±2,±5)
9.5.4.17.
Answer.
(±6,±2)
9.5.4.19.
Answer.
(0,4), (2,0)

9.6 Chapter 9 Summary and Review
9.6.3 Chapter 9 Review Problems

9.6.3.2.

Answer.
perpendicular

9.6.3.9.

Answer.

9.6.3.10.

Answer.

9.6.3.11.

Answer.

9.6.3.12.

Answer.

9.6.3.13.

Answer.

9.6.3.14.

Answer.

9.6.3.15.

Answer.

9.6.3.16.

Answer.

9.6.3.17.

Answer.

9.6.3.18.

Answer.

9.6.3.27.

Answer.
  1. (y4)26(x+2)24=1
  2. Hyperbola: center (2,4), transverse axis vertical, a=2, b=6

9.6.3.28.

Answer.
  1. (x4)24(y+3)29=1
  2. Hyperbola: center (4,3), transverse axis horizontal, a=2, b=3

9.6.3.29.

Answer.
  1. x25(y3)210=1
  2. Hyperbola: center (0,3), transverse axis horizontal, a=5, b=10

9.6.3.30.

Answer.
  1. y23(x4)212=1
  2. Hyperbola: center (4,0), transverse axis vertical, a=23, b=310

9.6.3.33.

Answer.
(x+4)2+(y3)2=20

9.6.3.34.

Answer.
(x+2)2+(y4)2=13

9.6.3.35.

Answer.
(x+1)216+(y4)24=1

9.6.3.36.

Answer.
(x3)24+(y1)225=1

9.6.3.37.

Answer.
(x2)216(y+3)29=1

9.6.3.38.

Answer.
(x+3)2(y1)29=1

9.6.3.41.

Answer.
(1,2), (1,2), (23,13), (23,13)

9.6.3.42.

Answer.
(342,342), (342,342)

9.6.3.43.

Answer.
Moia: 45 mph, Fran: 50 mph

9.6.3.44.

Answer.
12 in by 1 in

9.6.3.45.

Answer.
7 cm by 10 cm

9.6.3.46.

Answer.
7 ft by 2 ft

9.6.3.47.

Answer.
Morning train: 20 mph, evening train: 30 mph

9.6.3.48.

Answer.
Amount: $800, rate: 4%

10 Logarithmic Functions
10.1 Logarithmic Functions
10.1.7 Problem Set 10.1

Warm Up

10.1.7.7.
Answer.
log10(xyz3)

Applications

10.1.7.29.
Answer.
  1. data points and log curve
  2. The graph resembles a logarithmic function. The function is close to the points but appears too steep at first and not steep enough after n=15. Overall, it is a good fit.
  3. f grows (more and more slowly) without bound. f will eventually exceed 100 per cent, but no one can forget more than 100% of what is learned.

10.2 Logarithmic Scales
10.2.7 Problem Set 10.2

Applications

10.2.7.19.
Answer.
1, 80, 330, 1600, 7000, 4×107
10.2.7.21.
Answer.
Proxima Centauri: 15.5; Barnard: 13.2; Sirius: 1.4; Vega: 0.6; Arcturus: 0.4; Antares: 4.7; Betelgeuse: 7.2
10.2.7.23.
Answer.
10.2.7.25.
Answer.
103.42512
10.2.7.27.
Answer.
A: a45, p7.4%; B: a400, p15%; C: a6000, p50%; D: a13000, p45%

10.3 The Natural Base
10.3.7 Homework 10.3

Skills Practice

10.3.7.1.
Answer.
x 10 5 0 5 10 15 20
f(x) 0.135 0.368 1 2.718 7.389 20.086 54.598
growth
10.3.7.3.
Answer.
x 10 5 0 5 10 15 20
f(x) 20.086 4.482 1 0.223 0.05 0.011 0.00248
decay
10.3.7.11.
Answer.
P(t)=20(e0.4)t201.492t; increasing; initial value 20
10.3.7.13.
Answer.
P(t)=6500(e2.5)t65000.082t; decreasing; initial value 6500
10.3.7.15.
Answer.
  1. x 0 0.5 1 1.5 2 2.5
    ex 1 1.6487 2.7183 4.4817 7.3891 12.1825
  2. Each ratio is e0.51.6487: Increasing x-values by a constant Δx=0.5 corresponds to multiplying the y-values of the exponential function by a constant factor of eΔx.
10.3.7.17.
Answer.
  1. x 0 0.6931 1.3863 2.0794 2.7726 3.4657 4.1589
    ex 1 2 4 8 16 32 64
  2. Each difference in x-values is approximately ln20.6931: Increasing x-values by a constant Δx=ln2 corresponds to multiplying the y-values of the exponential function by a constant factor of eΔx=eln2=2. That is, each function value is approximately equal to double the previous one.
10.3.7.27.
Answer.
t=1kln(y)
10.3.7.29.
Answer.
t=ln(kky)
10.3.7.31.
Answer.
k=eT/T010
10.3.7.33.
Answer.
  1. n 0.39 3.9 39 390
    ln(n) 0.942 1.361 3.664 5.966
  2. Each difference in function values is approximately ln102.303: Multiplying x-values by a constant factor of 10 corresponds to adding a constant value of ln10 to the y-values of the natural log function.
10.3.7.35.
Answer.
  1. n 2 4 8 16
    lnn 0.693 1.386 2.079 2.773
  2. Each quotient equals k, where n=2k. Because ln(n)=ln(2k)=kln(2), k=ln(n)ln(2).

Applications

10.4 Chapter 10 Summary and Review
10.4.3 Chapter 10 Review Problems

10.4.3.31.

Answer.
t=1kln(y612)

10.4.3.35.

Answer.
P(t)=750(1.3771)t

10.4.3.37.

Answer.
N(t)=600e0.9163t

10.4.3.39.

Answer.

10.4.3.41.

Answer.
Order 3: 17,000; Order 4: 5000; Order 8: 40; Order 9: 11