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Solve each equation.
5x 2x − 4
1. 3 = 27
SOLUTION:
Use the Property of Equality for Exponential
Functions.
ANSWER:
12
2y − 3 y + 1
2. 16 = 4
SOLUTION:
Use the Property of Equality for Exponential
Functions.
ANSWER:
7-2 Solving Exponential Equations and Inequalities
6x x − 2
Solve each equation. 3. 2 = 32
5x 2x − 4
1. 3 = 27
SOLUTION:
SOLUTION:
Use the Property of Equality for Exponential
Use the Property of Equality for Exponential Functions.
Functions.
ANSWER:
ANSWER: −10
12 x + 5 8x − 6
4. 49 = 7
2y − 3 y + 1
2. 16 = 4
SOLUTION:
SOLUTION:
Use the Property of Equality for Exponential
Use the Property of Equality for Exponential Functions.
Functions.
ANSWER:
ANSWER:
SCIENCE
5. Mitosis is a process in which one cell
6x x − 2 divides into two. The Escherichia coli is one of the
3. 2 = 32 fastest growing bacteria. It can reproduce itself in 15
minutes.
SOLUTION: a. Write an exponential function to represent the
number of cells c after t minutes.
b. If you begin with one Escherichia coli cell, how
many cells will there be in one hour?
SOLUTION:
Use the Property of Equality for Exponential a.
Functions. The exponential function that represent the number
of cells after t minutes is .
eSolutions Manual - Powered by Cognero b. Page1
Substitute 1 for t in the function and solve for c.
ANSWER:
−10
ANSWER:
x + 5 8x − 6
4. 49 = 7 a.
SOLUTION: b. 16 cells
6. A certificate of deposit (CD) pays 2.25% annual
interest compounded biweekly. If you deposit $500
into this CD, what will the balance be after 6 years?
SOLUTION:
Use the Property of Equality for Exponential Use the compound interest formula.
Functions. Substitute $500 for P, 0.0225 for r, 26 for n and 6 for
t and simplify.
ANSWER:
ANSWER:
SCIENCE $572.23
5. Mitosis is a process in which one cell
divides into two. The Escherichia coli is one of the Solve each inequality.
fastest growing bacteria. It can reproduce itself in 15 2x + 6 2x – 4
minutes. 7. 4 ≤64
a. Write an exponential function to represent the
number of cells c after t minutes. SOLUTION:
b. If you begin with one Escherichia coli cell, how
many cells will there be in one hour?
SOLUTION:
a.
The exponential function that represent the number Use the Property of Inequality for Exponential
of cells after t minutes is . Functions.
b.
Substitute 1 for t in the function and solve for c.
ANSWER:
ANSWER:
a. x ≥ 4.5
b. 16 cells
6. A certificate of deposit (CD) pays 2.25% annual 8.
interest compounded biweekly. If you deposit $500
into this CD, what will the balance be after 6 years?
SOLUTION:
SOLUTION:
Use the compound interest formula.
Substitute $500 for P, 0.0225 for r, 26 for n and 6 for
t and simplify.
Use the Property of Inequality for Exponential
Functions.
ANSWER:
ANSWER:
$572.23
Solve each inequality. Solve each equation.
4x + 2
2x + 6 2x – 4 9. 8 = 64
7. 4 ≤64
SOLUTION:
SOLUTION:
Use the Property of Equality for Exponential
Functions.
Use the Property of Inequality for Exponential
Functions.
ANSWER:
0
x − 6
10. 5 = 125
ANSWER:
x ≥ 4.5
SOLUTION:
8.
Use the Property of Equality for Exponential
SOLUTION: Functions.
ANSWER:
9
Use the Property of Inequality for Exponential
Functions. a + 2 3a + 1
11. 81 = 3
SOLUTION:
ANSWER:
Solve each equation.
4x + 2 Use the Property of Equality for Exponential
9. 8 = 64 Functions.
SOLUTION:
Use the Property of Equality for Exponential ANSWER:
−7
Functions.
b + 2 2 − 2b
12. 256 = 4
SOLUTION:
ANSWER:
0
x − 6
10. 5 = 125
Use the Property of Equality for Exponential
SOLUTION: Functions.
Use the Property of Equality for Exponential
Functions.
ANSWER:
−1
3c + 1 3c − 1
13. 9 = 27
ANSWER:
9 SOLUTION:
a + 2 3a + 1
11. 81 = 3
SOLUTION:
Use the Property of Equality for Exponential
Functions.
Use the Property of Equality for Exponential
Functions.
ANSWER:
ANSWER:
−7
2y + 4 y + 1
b + 2 2 − 2b 14. 8 = 16
12. 256 = 4
SOLUTION:
SOLUTION:
Use the Property of Equality for Exponential Use the Property of Equality for Exponential
Functions. Functions.
ANSWER:
ANSWER:
−4
−1
3c + 1 3c − 1
13. 9 = 27 15. CCSS MODELINGIn 2009, My-Lien received
$10,000 from her grandmother. Her parents invested
SOLUTION: all of the money, and by 2021, the amount will have
grown to $16,960.
a. Write an exponential function that could be used to
model the money y. Write the function in terms of x,
the number of years since 2009.
b. Assume that the amount of money continues to
Use the Property of Equality for Exponential grow at the same rate. What would be the balance in
Functions. the account in 2031?
SOLUTION:
a.
Substitute 16780 for y 10000 for a and 12 for x in the
exponential function and simplify.
ANSWER:
2y + 4 y + 1
14. 8 = 16
The exponential function that models the situation
SOLUTION: is .
b.
Substitute 22 for x in the modeled function and solve
for y.
Use the Property of Equality for Exponential
Functions.
ANSWER:
x
a
. y = 10,000(1.045)
b
. about $26,336.52
ANSWER: Write an exponential function for the graph that
−4
passes through the given points.
16. (0, 6.4) and (3, 100)
15. CCSS MODELINGIn 2009, My-Lien received
$10,000 from her grandmother. Her parents invested
all of the money, and by 2021, the amount will have SOLUTION:
grown to $16,960. Substitute 100 for y and 6.4 for a and 3 for x into an
a. Write an exponential function that could be used to exponential function and determine the value of b.
model the money y. Write the function in terms of x,
the number of years since 2009.
b. Assume that the amount of money continues to
grow at the same rate. What would be the balance in
the account in 2031?
SOLUTION:
a.
Substitute 16780 for y 10000 for a and 12 for x in the An exponential function that passes through the given
exponential function and simplify. points is .
ANSWER:
x
y = 6.4(2.5)
17. (0, 256) and (4, 81)
The exponential function that models the situation SOLUTION:
is . Substitute 81 for y and 256 for a and 4 for x into an
exponential function and determine the value of b.
b.
Substitute 22 for x in the modeled function and solve
for y.
ANSWER: An exponential function that passes through the given
x
a .
. y = 10,000(1.045) points is
b
. about $26,336.52
Write an exponential function for the graph that ANSWER:
x
y = 256(0.75)
passes through the given points.
16. (0, 6.4) and (3, 100) 18. (0, 128) and (5, 371,293)
SOLUTION:
Substitute 100 for y and 6.4 for a and 3 for x into an SOLUTION:
exponential function and determine the value of b. Substitute 371293 for y and 128 for a and 5 for x into
an exponential function and determine the value of b.
An exponential function that passes through the given An exponential function that passes through the given
points is . points is .
ANSWER:
x ANSWER:
y = 6.4(2.5) x
y = 128(4.926)
17. (0, 256) and (4, 81) 19. (0, 144), and (4, 21,609)
SOLUTION:
SOLUTION:
Substitute 81 for y and 256 for a and 4 for x into an Substitute 21609 for y and 144 for a and 4 for x into
exponential function and determine the value of b. an exponential function and determine the value of b.
An exponential function that passes through the given An exponential function that passes through the given
points is . points is .
ANSWER:
ANSWER:
x x
y = 256(0.75) y = 144(3.5)
18. (0, 128) and (5, 371,293) 20. Find the balance of an account after 7 years if $700
is deposited into an account paying 4.3% interest
SOLUTION: compounded monthly.
Substitute 371293 for y and 128 for a and 5 for x into
an exponential function and determine the value of b. SOLUTION:
Use the compound interest formula.
Substitute $700 for P, 0.043 for r, 12 for n and 7 for t
and simplify.
An exponential function that passes through the given
points is .
ANSWER:
ANSWER: $945.34
x
y = 128(4.926) 21. Determine how much is in a retirement account after
19. (0, 144), and (4, 21,609) 20 years if $5000 was invested at 6.05% interest
compounded weekly.
SOLUTION:
Substitute 21609 for y and 144 for a and 4 for x into SOLUTION:
an exponential function and determine the value of b. Use the compound interest formula.
Substitute $5000 for P, 0.0605 for r, 52 for n and 20
for t and simplify.
An exponential function that passes through the given
points is . ANSWER:
$16,755.63
ANSWER: 22. A savings account offers 0.7% interest compounded
x
y = 144(3.5) bimonthly. If $110 is deposited in this account, what
20. Find the balance of an account after 7 years if $700 will the balance be after 15 years?
is deposited into an account paying 4.3% interest SOLUTION:
compounded monthly. Use the compound interest formula.
Substitute $110 for P, 0.007 for r, 6 for n and 15 for t
SOLUTION: and simplify.
Use the compound interest formula.
Substitute $700 for P, 0.043 for r, 12 for n and 7 for t
and simplify.
ANSWER:
$122.17
ANSWER: 23. A college savings account pays 13.2% annual
$945.34 interest compounded semiannually. What is the
21. Determine how much is in a retirement account after balance of an account after 12 years if $21,000 was
20 years if $5000 was invested at 6.05% interest initially deposited?
compounded weekly.
SOLUTION:
Use the compound interest formula.
SOLUTION: Substitute $21,000 for P, 0.132 for r, 2 for n and 12
Use the compound interest formula. for t and simplify.
Substitute $5000 for P, 0.0605 for r, 52 for n and 20
for t and simplify.
ANSWER:
$97,362.61
ANSWER:
$16,755.63 Solve each inequality.
22. A savings account offers 0.7% interest compounded 24.
bimonthly. If $110 is deposited in this account, what
will the balance be after 15 years? SOLUTION:
SOLUTION:
Use the compound interest formula.
Substitute $110 for P, 0.007 for r, 6 for n and 15 for t Use the Property of Inequality for Exponential
and simplify. Functions.
ANSWER:
ANSWER: 25.
$122.17
SOLUTION:
23. A college savings account pays 13.2% annual
interest compounded semiannually. What is the
balance of an account after 12 years if $21,000 was
initially deposited? Use the Property of Inequality for Exponential
Functions.
SOLUTION:
Use the compound interest formula.
Substitute $21,000 for P, 0.132 for r, 2 for n and 12
for t and simplify.
ANSWER:
26.
ANSWER:
$97,362.61
SOLUTION:
Solve each inequality.
24.
SOLUTION:
Use the Property of Inequality for Exponential Use the Property of Inequality for Exponential
Functions. Functions.
ANSWER:
ANSWER:
25.
SOLUTION:
27.
Use the Property of Inequality for Exponential SOLUTION:
Functions.
Use the Property of Inequality for Exponential
Functions.
ANSWER:
26.
ANSWER:
SOLUTION:
28.
SOLUTION:
Use the Property of Inequality for Exponential
Functions.
Use the Property of Inequality for Exponential
Functions.
ANSWER:
ANSWER:
27.
SOLUTION:
29.
SOLUTION:
Use the Property of Inequality for Exponential
Functions.
Use the Property of Inequality for Exponential
Functions.
ANSWER:
28.
SOLUTION:
ANSWER:
30. SCIENCEA mug of hot chocolate is 90°C at time t
= 0. It is surrounded by air at a constant temperature
of 20°C. If stirred steadily, its temperature in Celsius
−t
after t minutes will be y(t) = 20 + 70(1.071) .
a. Find the temperature of the hot chocolate after 15
Use the Property of Inequality for Exponential minutes.
Functions. b. Find the temperature of the hot chocolate after 30
minutes.
c.
The optimum drinking temperature is 60°C. Will
the mug of hot chocolate be at or below this
temperature after 10 minutes?
SOLUTION:
ANSWER: a.
Substitute 15 for t in the equation and simplify.
29.
b.
SOLUTION: Substitute 30 for t in the equation and simplify.
c.
Substitute 10 for t in the equation and simplify.
So, temperature of the hot chocolate will be below
Use the Property of Inequality for Exponential
Functions. 60°C after 10 minutes.
ANSWER:
a
. 45.02° C
b
. 28.94° C
c. below
31. ANIMALSStudies show that an animal will defend
a territory, with area in square yards, that is directly
ANSWER: proportional to the 1.31 power of the animal’s weight
in pounds.
a. If a 45-pound beaver will defend 170 square yards,
30. SCIENCEA mug of hot chocolate is 90°C at time t write an equation for the area a defended by a
= 0. It is surrounded by air at a constant temperature beaver weighing w pounds.
of 20°C. If stirred steadily, its temperature in Celsius b. Scientists believe that thousands of years ago, the
−t beaver’s ancestors were 11 feet long and weighed
after t minutes will be y(t) = 20 + 70(1.071) . 430 pounds. Use your equation to determine the area
a. Find the temperature of the hot chocolate after 15 defended by these animals.
minutes.
b. Find the temperature of the hot chocolate after 30
minutes. SOLUTION:
a.
c.
The optimum drinking temperature is 60°C. Will Substitute 170 for y, 45 for b, and 1.31 for x in the
the mug of hot chocolate be at or below this exponential function.
temperature after 10 minutes?
SOLUTION:
a.
Substitute 15 for t in the equation and simplify.
The equation for the area a defended by a beaver
b. weighting w pounds is
Substitute 30 for t in the equation and simplify. b.
Substitute 430 for w in the equation and solve for y.
c.
Substitute 10 for t in the equation and simplify.
ANSWER:
So, temperature of the hot chocolate will be below a 1.31
. a = 1.16w
2
60°C after 10 minutes. b
. about 3268 yd
ANSWER: Solve each equation.
a
. 45.02° C
b
. 28.94° C
c 32.
. below
SOLUTION:
31. ANIMALSStudies show that an animal will defend
a territory, with area in square yards, that is directly
proportional to the 1.31 power of the animal’s weight
in pounds.
a. If a 45-pound beaver will defend 170 square yards,
write an equation for the area a defended by a
beaver weighing w pounds.
b. Scientists believe that thousands of years ago, the Use the Property of Equality for Exponential
beaver’s ancestors were 11 feet long and weighed Functions.
430 pounds. Use your equation to determine the area
defended by these animals.
SOLUTION:
a.
Substitute 170 for y, 45 for b, and 1.31 for x in the
exponential function.
ANSWER:
The equation for the area a defended by a beaver 33.
weighting w pounds is
b. SOLUTION:
Substitute 430 for w in the equation and solve for y.
ANSWER: Use the Property of Equality for Exponential
a 1.31
. a = 1.16w Functions.
b 2
. about 3268 yd
Solve each equation.
32.
SOLUTION:
ANSWER:
34.
Use the Property of Equality for Exponential
Functions. SOLUTION:
Use the Property of Equality for Exponential
Functions.
ANSWER:
ANSWER:
33. −6
SOLUTION:
35.
SOLUTION:
Use the Property of Equality for Exponential
Functions.
Use the Property of Equality for Exponential
Functions.
ANSWER:
ANSWER:
34.
SOLUTION:
36.
SOLUTION:
Use the Property of Equality for Exponential
Functions.
ANSWER:
−6
Use the Property of Equality for Exponential
Functions.
35.
SOLUTION:
ANSWER:
Use the Property of Equality for Exponential
Functions. 37.
SOLUTION:
ANSWER:
Use the Property of Equality for Exponential
Functions.
36.
SOLUTION:
ANSWER:
1
38. CCSS MODELINGIn 1950, the world population
was about 2.556 billion. By 1980, it had increased to
about 4.458 billion.
a. x
Write an exponential function of the form y = ab
that could be used to model the world population y in
billions for 1950 to 1980. Write the equation in terms
Use the Property of Equality for Exponential of x, the number of years since 1950. (Round the
Functions. value of b to the nearest ten-thousandth.)
b. Suppose the population continued to grow at that
rate. Estimate the population in 2000.
c. In 2000, the population of the world was about
6.08 billion. Compare your estimate to the actual
population.
d. Use the equation you wrote in Part a to estimate
the world population in the year 2020. How accurate
ANSWER: do you think the estimate is? Explain your reasoning.
SOLUTION:
a.
Substitute 4.458 for y, 2.556 for a, and 30 for x in the
37. exponential function and solve for b.
SOLUTION:
The exponential function that model the situation is
.
Use the Property of Equality for Exponential b.
Functions. Substitute 50 for x in the equation and simplify.
c. The prediction was about 375 million greater than
ANSWER: the actual population.
1
d.
Substitute 70 for x in the equation and simplify.
38. CCSS MODELINGIn 1950, the world population
was about 2.556 billion. By 1980, it had increased to
about 4.458 billion.
a. x
Write an exponential function of the form y = ab
that could be used to model the world population y in
billions for 1950 to 1980. Write the equation in terms Because the prediction for 2000 was greater than the
of x, the number of years since 1950. (Round the actual population, this prediction for 2020 is probably
value of b to the nearest ten-thousandth.) even higher than the actual population will be at the
b. Suppose the population continued to grow at that time.
rate. Estimate the population in 2000.
ANSWER:
c. In 2000, the population of the world was about
x
6.08 billion. Compare your estimate to the actual a
population. . y = 2.556(1.0187)
b
d. Use the equation you wrote in Part a to estimate . 6.455 billion
the world population in the year 2020. How accurate c. The prediction was about 375 million greater than
do you think the estimate is? Explain your reasoning. the actual.
d
. About 9.3498 billion; because the prediction for
2000 was greater than the actual population, this
SOLUTION: prediction is probably even higher than the actual
a. population will be at the time.
Substitute 4.458 for y, 2.556 for a, and 30 for x in the
exponential function and solve for b. 39. TREES The diameter of the base of a tree trunk in
centimeters varies directly with the power of its
height in meters.
a. A young sequoia tree is 6 meters tall, and the
diameter of its base is 19.1 centimeters. Use this
The exponential function that model the situation is information to write an equation for the diameter d of
. the base of a sequoia tree if its height is h meters
high
b. b. The General Sherman Tree in Sequoia National
Substitute 50 for x in the equation and simplify. Park, California, is approximately 84 meters tall.
Find the diameter of the General Sherman Tree at its
base.
SOLUTION:
c. The prediction was about 375 million greater than a.
the actual population. The equation that represent the situation is
d.
Substitute 70 for x in the equation and simplify. .
b.
Substitute 84 for h in the equation and solve for d.
Because the prediction for 2000 was greater than the
actual population, this prediction for 2020 is probably
even higher than the actual population will be at the The diameter of the General Sherman Tree at its
time. base is about 1001 cm.
ANSWER:
ANSWER:
x
a
. y = 2.556(1.0187) a.
b
. 6.455 billion b
c. The prediction was about 375 million greater than . about 1001 cm
the actual.
d 40. FINANCIAL LITERACYMrs. Jackson has two
. About 9.3498 billion; because the prediction for different retirement investment plans from which to
2000 was greater than the actual population, this choose.
prediction is probably even higher than the actual a. Write equations for Option A and Option B given
population will be at the time. the minimum deposits.
39. TREES The diameter of the base of a tree trunk in b. Draw a graph to show the balances for each
investment option after t years.
centimeters varies directly with the power of its c. Explain whether Option A or Option B is the
height in meters. better investment choice.
a. A young sequoia tree is 6 meters tall, and the
diameter of its base is 19.1 centimeters. Use this
information to write an equation for the diameter d of
the base of a sequoia tree if its height is h meters
high
b. The General Sherman Tree in Sequoia National
Park, California, is approximately 84 meters tall.
Find the diameter of the General Sherman Tree at its
base.
SOLUTION:
a.
SOLUTION: Use the compound interest formula.
a. The equation that represents Option A
The equation that represent the situation is
. is .
b. The equation that represents Option B
Substitute 84 for h in the equation and solve for d.
is
b.
The graph that shows the balances for each
investment option after t years:
The diameter of the General Sherman Tree at its
base is about 1001 cm.
ANSWER:
a.
b
. about 1001 cm
40. FINANCIAL LITERACYMrs. Jackson has two
different retirement investment plans from which to
choose.
a. Write equations for Option A and Option B given
the minimum deposits.
b. Draw a graph to show the balances for each
investment option after t years.
c.
c. Explain whether Option A or Option B is the During the first 22 years, Option B is the better
better investment choice. choice because the total is greater than that of
Option A. However, after about 22 years, the
balance of Option A exceeds that of Option B, so
Option A is the better choice.
ANSWER:
a.
b.
SOLUTION:
a.
Use the compound interest formula.
The equation that represents Option A
is .
The equation that represents Option B
is
b.
The graph that shows the balances for each
investment option after t years:
Sample answer:
c. During the first 22 years, Option
B is the better choice because the total is greater
than that of Option A. However, after about 22
years, the balance of Option A exceeds that of
Option B, so Option A is the better choice.
41. MULTIPLE REPRESENTATIONSIn this
problem, you will explore the rapid increase of an
exponential function. A large sheet of paper is cut in
half, and one of the resulting pieces is placed on top
of the other. Then the pieces in the stack are cut in
half and placed on top of each other. Suppose this
procedure is repeated several times.
a. CONCRETE
Perform this activity and count the
number of sheets in the stack after the first cut. How
many pieces will there be after the second cut? How
c. many pieces after the third cut? How many pieces
During the first 22 years, Option B is the better after the fourth cut?
choice because the total is greater than that of b. TABULAR
Option A. However, after about 22 years, the Record your results in a table.
c. SYMBOLIC
balance of Option A exceeds that of Option B, so Use the pattern in the table to write
Option A is the better choice. an equation for the number of pieces in the stack
after x cuts.
ANSWER: d. ANALYTICAL
The thickness of ordinary paper
a. is about 0.003 inch. Write an equation for the
thickness of the stack of paper after x cuts.
e.ANALYTICAL
b. How thick will the stack of
paper be after 30 cuts?
SOLUTION:
a.
There will be 2, 4, 8, 16 pieces after the first, second,
third and fourth cut respectively.
b.
c.
Sample answer: The equation that represent the situation is
c. During the first 22 years, Option d.
B is the better choice because the total is greater Substitute 0.003 for a and 2 for b in the exponential
than that of Option A. However, after about 22 function.
years, the balance of Option A exceeds that of
Option B, so Option A is the better choice.
41. MULTIPLE REPRESENTATIONSIn this
problem, you will explore the rapid increase of an e.
exponential function. A large sheet of paper is cut in Substitute 30 for x in the equation and
half, and one of the resulting pieces is placed on top simplify.
of the other. Then the pieces in the stack are cut in
half and placed on top of each other. Suppose this
procedure is repeated several times.
a. CONCRETE
Perform this activity and count the
number of sheets in the stack after the first cut. How The thickness of the stack of paper after 30 cuts is
many pieces will there be after the second cut? How about 3221225.47 in.
many pieces after the third cut? How many pieces
after the fourth cut?
b. TABULAR ANSWER:
Record your results in a table. a. 2, 4, 8, 16
c. SYMBOLIC
Use the pattern in the table to write b
an equation for the number of pieces in the stack .
after x cuts.
d. ANALYTICAL
The thickness of ordinary paper
is about 0.003 inch. Write an equation for the
thickness of the stack of paper after x cuts.
e.ANALYTICAL
How thick will the stack of
paper be after 30 cuts? x
c. y = 2
x
SOLUTION: d
. y = 0.003(2)
a. e. about 3,221,225.47 in.
There will be 2, 4, 8, 16 pieces after the first, second,
third and fourth cut respectively. 42. WRITING IN MATHIn a problem about
b. compound interest, describe what happens as the
compounding period becomes more frequent while
the principal and overall time remain the same.
SOLUTION:
Sample answer: The more frequently interest is
compounded, the higher the account balance
becomes.
c.
The equation that represent the situation is
d. ANSWER:
Substitute 0.003 for a and 2 for b in the exponential Sample answer: The more frequently interest is
function. compounded, the higher the account balance
becomes.
x −
ERROR ANALYSIS
43. Beth and Liz are solving 6
3 −x − 1
e. > 36 . Is either of them correct? Explain your
Substitute 30 for x in the equation and reasoning.
simplify.
The thickness of the stack of paper after 30 cuts is
about 3221225.47 in.
ANSWER:
a. 2, 4, 8, 16
b
.
x
c. y = 2
x
d
. y = 0.003(2)
e. about 3,221,225.47 in.
42. WRITING IN MATHIn a problem about
compound interest, describe what happens as the
compounding period becomes more frequent while
the principal and overall time remain the same.
SOLUTION:
Sample answer: Beth; Liz added the exponents
SOLUTION: instead of multiplying them when taking the power of
Sample answer: The more frequently interest is
compounded, the higher the account balance a power.
becomes.
ANSWER:
Sample answer: Beth; Liz added the exponents
ANSWER: instead of multiplying them when taking the power of
Sample answer: The more frequently interest is
compounded, the higher the account balance a power.
becomes. 18 18 18
44. CHALLENGESolve for x: 16 + 16 + 16 +
x − 18 18 x
ERROR ANALYSIS 16 + 16 = 4 .
43. Beth and Liz are solving 6
3 > 36−x − 1. Is either of them correct? Explain your
reasoning. SOLUTION:
ANSWER:
37.1610
45. OPEN ENDEDWhat would be a more beneficial
change to a 5-year loan at 8% interest compounded
monthly: reducing the term to 4 years or reducing the
interest rate to 6.5%?
SOLUTION:
Reducing the term will be more beneficial. The
multiplier is 1.3756 for the 4-year and 1.3828 for the
6.5%.
ANSWER:
Reducing the term will be more beneficial. The
multiplier is 1.3756 for the 4-year and 1.3828 for the
6.5%.
SOLUTION:
Sample answer: Beth; Liz added the exponents
CCSS ARGUMENTS
instead of multiplying them when taking the power of 46. Determine whether the
following statements are sometimes, always, or
a power. never true. Explain your reasoning.
a. x 20x
2 > −8 for all values of x.
ANSWER:
Sample answer: Beth; Liz added the exponents b. The graph of an exponential growth equation is
instead of multiplying them when taking the power of increasing.
a power. c. The graph of an exponential decay equation is
increasing.
18 18 18
44. CHALLENGESolve for x: 16 + 16 + 16 +
18 18 x SOLUTION:
16 + 16 = 4 . a. x 20x
Always; 2 will always be positive, and −8 will
SOLUTION: always be negative.
b. Always; by definition the graph will always be
increasing even if it is a small increase.
c. Never; by definition the graph will always be
decreasing even if it is a small decrease.
ANSWER:
a x 20x
. Always; 2 will always be positive, and 8 will
−
always be negative.
b
. Always; by definition the graph will always be
increasing even if it is a small increase.
c. Never; by definition the graph will always be
ANSWER: decreasing even if it is a small decrease.
37.1610
OPEN ENDEDWrite an exponential inequality with
47.
45. OPEN ENDEDWhat would be a more beneficial a solution of x 2.
change to a 5-year loan at 8% interest compounded ≤
monthly: reducing the term to 4 years or reducing the
interest rate to 6.5%? SOLUTION:
x 2
Sample answer: 4 4
≤
SOLUTION:
Reducing the term will be more beneficial. The ANSWER:
multiplier is 1.3756 for the 4-year and 1.3828 for the x 2
Sample answer: 4 4
≤
6.5%.
2x x + 1 2x + 2 4x + 1
PROOFShow that 27 · 81 = 3 · 9 .
48.
ANSWER:
Reducing the term will be more beneficial. The
multiplier is 1.3756 for the 4-year and 1.3828 for the SOLUTION:
6.5%.
CCSS ARGUMENTS
46. Determine whether the
following statements are sometimes, always, or
never true. Explain your reasoning.
a. x 20x
2 > −8 for all values of x.
b. The graph of an exponential growth equation is
increasing.
c. The graph of an exponential decay equation is
increasing.
SOLUTION: ANSWER:
a. x 20x
Always; 2 will always be positive, and −8 will
always be negative.
b. Always; by definition the graph will always be
increasing even if it is a small increase.
c. Never; by definition the graph will always be
decreasing even if it is a small decrease.
ANSWER:
x 20x
WRITING IN MATHIf you were given the initial
a 49.
. Always; 2 will always be positive, and 8 will
− and final amounts of a radioactive substance and the
always be negative. amount of time that passes, how would you
b
. Always; by definition the graph will always be determine the rate at which the amount was
increasing even if it is a small increase. increasing or decreasing in order to write an
c. Never; by definition the graph will always be equation?
decreasing even if it is a small decrease.
SOLUTION:
OPEN ENDEDWrite an exponential inequality with
47. Sample answer: Divide the final amount by the initial
a solution of x 2.
≤ amount. If n is the number of time intervals that pass,
take the nth root of the answer.
SOLUTION:
x 2
Sample answer: 4 4
≤ ANSWER:
Sample answer: Divide the final amount by the initial
ANSWER: amount. If n is the number of time intervals that pass,
x 2
Sample answer: 4 4 take the nth root of the answer.
≤
2x x + 1 2x + 2 4x + 1 −4 =
50. 3 × 10
PROOFShow that 27 · 81 = 3 · 9 .
48.
A 30,000
−
SOLUTION: B 0.0003
C 120
−
D 0.00003
SOLUTION:
B is the correct option.
ANSWER:
ANSWER:
B
51. Which of the following could not be a solution to 5 −
3x < 3?
−
F2.5
G3
H 3.5
WRITING IN MATHIf you were given the initial J
49. 4
and final amounts of a radioactive substance and the
amount of time that passes, how would you SOLUTION:
determine the rate at which the amount was Check the inequality by substituting 2.5 for x.
increasing or decreasing in order to write an
equation?
SOLUTION:
Sample answer: Divide the final amount by the initial
amount. If n is the number of time intervals that pass, So, F is the correct option.
take the nth root of the answer.
ANSWER:
ANSWER: F
Sample answer: Divide the final amount by the initial
GRIDDED RESPONSEThe three angles of a
amount. If n is the number of time intervals that pass, 52.
take the nth root of the answer. triangle are 3x, x + 10, and 2x − 40. Find the measure
of the smallest angle in the triangle.
−4 =
50. 3 × 10
A 30,000 SOLUTION:
− Sum of the three angles in a triangle is 180 .
B 0.0003 º
C 120
−
D 0.00003
SOLUTION:
B is the correct option.
The measure of the smallest angle in the triangle is
ANSWER: 30 .
º
B
ANSWER:
51. Which of the following could not be a solution to 5 − 30
3x < 3?
−
SAT/ACTWhich of the following is equivalent to
53.
F2.5
(x)(x)(x)(x) for all x?
G3
H 3.5
J A x + 4
4 B 4
x
SOLUTION: C 2x2
Check the inequality by substituting 2.5 for x. D 4x2
4
E
x
SOLUTION:
So, F is the correct option.
ANSWER:
F E is the correct choice.
ANSWER:
GRIDDED RESPONSEThe three angles of a
52. E
triangle are 3x, x + 10, and 2x 40. Find the measure
−
of the smallest angle in the triangle. Graph each function.
x
SOLUTION: y = 2(3)
Sum of the three angles in a triangle is 180 . 54.
º
SOLUTION:
Make a table of values. Then plot the points and
sketch the graph.
The measure of the smallest angle in the triangle is
30 .
º
ANSWER:
30
SAT/ACT Which of the following is equivalent to
53.
(x)(x)(x)(x) for all x?
A x + 4
B 4x
C 2x2
D 4x2 ANSWER:
E 4
x
SOLUTION:
E is the correct choice.
ANSWER:
E
x
y = 5(2)
Graph each function. 55.
x
y = 2(3)
54.
SOLUTION:
Make a table of values. Then plot the points and
SOLUTION: sketch the graph.
Make a table of values. Then plot the points and
sketch the graph.
ANSWER:
ANSWER:
x 56.
y = 5(2)
55.
SOLUTION:
SOLUTION: Make a table of values. Then plot the points and
Make a table of values. Then plot the points and sketch the graph.
sketch the graph.
ANSWER:
ANSWER:
Solve each equation.
57.
56.
SOLUTION:
SOLUTION:
Make a table of values. Then plot the points and
sketch the graph.
ANSWER:
4
58.
SOLUTION:
ANSWER:
18
59.
ANSWER:
SOLUTION:
ANSWER:
Solve each equation. 8.5
57.
60.
SOLUTION:
SOLUTION:
ANSWER:
4
The square root of x cannot be negative, so there is
no solution.
58.
ANSWER:
SOLUTION: no solution
61.
SOLUTION:
ANSWER:
18
59.
ANSWER:
SOLUTION: 5
62.
SOLUTION:
ANSWER:
8.5
60.
SOLUTION:
ANSWER:
20
−
63.
SOLUTION:
The square root of x cannot be negative, so there is
no solution.
ANSWER:
no solution
61.
ANSWER:
SOLUTION: 5
64.
SOLUTION:
ANSWER:
5
62.
SOLUTION:
ANSWER:
65.
SOLUTION:
ANSWER:
20
−
63.
SOLUTION:
ANSWER:
1
−
SALES A salesperson earns $10 an hour plus a 10%
66.
commission on sales. Write a function to describe the
salesperson’s income. If the salesperson wants to
earn $1000 in a 40-hour week, what should his sales
be?
ANSWER:
5
SOLUTION:
Let I be the income of the salesperson and m be his
64.
sales.
The function that represent the situation is
SOLUTION: .
Substitute 1000 for I in the equation and solve for m.
ANSWER:
I(m) = 400 + 0.1m; $6000
STATE FAIRA dairy makes three types of
67.
cheese cheddar, Monterey Jack, and Swiss and
— —
ANSWER: sells the cheese in three booths at the state fair. At
the beginning of one day, the first booth received x
pounds of each type of cheese. The second booth
received y pounds of each type of cheese, and the
third booth received z pounds of each type of cheese.
65. By the end of the day, the dairy had sold 131 pounds
SOLUTION: of cheddar, 291 pounds of Monterey Jack, and 232
pounds of Swiss. The table below shows the percent
of the cheese delivered in the morning that was sold
at each booth. How many pounds of cheddar cheese
did each booth receive in the morning?
ANSWER:
1
−
SOLUTION:
SALES A salesperson earns $10 an hour plus a 10% The system of equations that represent the situation:
66.
commission on sales. Write a function to describe the
salesperson s income. If the salesperson wants to
’
earn $1000 in a 40-hour week, what should his sales
be?
SOLUTION:
Let I be the income of the salesperson and m be his
Eliminate the variable x by using two pairs of
sales. equations.
The function that represent the situation is
.
Subtract (1) and (2).
Substitute 1000 for I in the equation and solve for m.
ANSWER: Multiply (2) by 3 and (3) by 4 and subtract both the
I(m) = 400 + 0.1m; $6000 equations.
STATE FAIRA dairy makes three types of
67.
cheese cheddar, Monterey Jack, and Swiss and
— —
sells the cheese in three booths at the state fair. At
the beginning of one day, the first booth received x
pounds of each type of cheese. The second booth
received y pounds of each type of cheese, and the
third booth received z pounds of each type of cheese. Solve the system of two equations:
By the end of the day, the dairy had sold 131 pounds
of cheddar, 291 pounds of Monterey Jack, and 232
pounds of Swiss. The table below shows the percent
of the cheese delivered in the morning that was sold
at each booth. How many pounds of cheddar cheese
did each booth receive in the morning?
Substitute z = 100 in the equation
SOLUTION:
The system of equations that represent the situation:
Substitute y = 150 and z = 100 in the (1) and solve
for x.
Eliminate the variable x by using two pairs of
equations.
Subtract (1) and (2).
Booth 1 has 190 lb; Booth 2 has 150 lb; Booth 3 has
100 lb.
ANSWER:
booth 1, 190 lb; booth 2, 150 lb; booth 3, 100 lb
Find [g h](x) and [h g](x).
Multiply (2) by 3 and (3) by 4 and subtract both the ◦ ◦
68. h(x) = 2x − 1
equations.
g(x) = 3x + 4
SOLUTION:
Solve the system of two equations:
ANSWER:
6x + 1; 6x + 7
2
Substitute z = 100 in the equation 69. h(x) = x + 2
g(x) = x − 3
SOLUTION:
Substitute y = 150 and z = 100 in the (1) and solve
for x.
ANSWER:
x2 1; x2 6x + 11
− −
h(x) = x2 + 1
70.
g(x) = 2x + 1
−
Booth 1 has 190 lb; Booth 2 has 150 lb; Booth 3 has
100 lb. SOLUTION:
ANSWER:
booth 1, 190 lb; booth 2, 150 lb; booth 3, 100 lb
Find [g h](x) and [h g](x).
◦ ◦
68. h(x) = 2x − 1
g(x) = 3x + 4
ANSWER:
SOLUTION:
2x2 1; 4x2 4x + 2
− − −
h(x) = 5x
71. −
g(x) = 3x − 5
SOLUTION:
ANSWER:
6x + 1; 6x + 7
h(x) = x2 + 2 ANSWER:
69. 15x 5; 15x + 25
− − −
g(x) = x − 3
3
SOLUTION: 72. h(x) = x
g(x) = x − 2
SOLUTION:
ANSWER:
ANSWER:
2 2 3 3 2
x x 6x + 12x 8
x 1; x 6x + 11 −2; − −
− −
h(x) = x2 + 1 73. h(x) = x + 4
70.
g(x) = 2x + 1 g(x) = | x |
−
SOLUTION:
SOLUTION:
ANSWER:
| x + 4 | ; | x | + 4
ANSWER:
2x2 1; 4x2 4x + 2
− − −
h(x) = 5x
71. −
g(x) = 3x − 5
SOLUTION:
ANSWER:
15x 5; 15x + 25
− − −
h(x) = x3
72.
g(x) = x − 2
SOLUTION:
ANSWER:
3 3 2
x x 6x + 12x 8
−2; − −
73. h(x) = x + 4
g(x) = | x |
SOLUTION:
ANSWER:
| x + 4 | ; | x | + 4
Solve each equation.
5x 2x − 4
1. 3 = 27
SOLUTION:
Use the Property of Equality for Exponential
Functions.
ANSWER:
12
2y − 3 y + 1
2. 16 = 4
SOLUTION:
Use the Property of Equality for Exponential
Functions.
ANSWER:
6x x − 2
3. 2 = 32
Solve each equation.
5x 2x − 4 SOLUTION:
1. 3 = 27
SOLUTION:
Use the Property of Equality for Exponential
Functions.
Use the Property of Equality for Exponential
Functions.
ANSWER:
−10
ANSWER: x + 5 8x − 6
12 4. 49 = 7
SOLUTION:
2y − 3 y + 1
2. 16 = 4
SOLUTION:
Use the Property of Equality for Exponential
Functions.
Use the Property of Equality for Exponential
Functions.
ANSWER:
ANSWER:
SCIENCE
5. Mitosis is a process in which one cell
divides into two. The Escherichia coli is one of the
fastest growing bacteria. It can reproduce itself in 15
6x x − 2 minutes.
3. 2 = 32 a. Write an exponential function to represent the
number of cells c after t minutes.
SOLUTION:
b. If you begin with one Escherichia coli cell, how
many cells will there be in one hour?
SOLUTION:
a.
The exponential function that represent the number
Use the Property of Equality for Exponential of cells after t minutes is .
Functions. b.
Substitute 1 for t in the function and solve for c.
ANSWER:
−10 ANSWER:
a.
x + 5 8x − 6
4. 49 = 7 b. 16 cells
SOLUTION: 6. A certificate of deposit (CD) pays 2.25% annual
interest compounded biweekly. If you deposit $500
into this CD, what will the balance be after 6 years?
SOLUTION:
Use the compound interest formula.
Use the Property of Equality for Exponential Substitute $500 for P, 0.0225 for r, 26 for n and 6 for
Functions. t and simplify.
ANSWER:
7-2 Solving Exponential Equations and Inequalities ANSWER:
$572.23
SCIENCE Solve each inequality.
5. Mitosis is a process in which one cell
divides into two. The Escherichia coli is one of the 2x + 6 2x – 4
fastest growing bacteria. It can reproduce itself in 15 7. 4 ≤64
minutes. SOLUTION:
a. Write an exponential function to represent the
number of cells c after t minutes.
b. If you begin with one Escherichia coli cell, how
many cells will there be in one hour?
SOLUTION:
a. Use the Property of Inequality for Exponential
The exponential function that represent the number Functions.
of cells after t minutes is .
b.
Substitute 1 for t in the function and solve for c.
ANSWER:
ANSWER: x ≥ 4.5
a.
b. 16 cells
8.
6. A certificate of deposit (CD) pays 2.25% annual
interest compounded biweekly. If you deposit $500 SOLUTION:
into this CD, what will the balance be after 6 years?
SOLUTION:
Use the compound interest formula.
Substitute $500 for P, 0.0225 for r, 26 for n and 6 for
t and simplify.
Use the Property of Inequality for Exponential
Functions.
ANSWER:
ANSWER: Solve each equation.
$572.23 4x + 2
9. 8 = 64
Solve each inequality.
2x + 6 2x – 4 SOLUTION:
7. 4 ≤64
SOLUTION:
Use the Property of Equality for Exponential
Functions.
Use the Property of Inequality for Exponential
eSolutions Manual - Powered by Cognero Page2
Functions.
ANSWER:
0
x − 6
10. 5 = 125
ANSWER: SOLUTION:
x ≥ 4.5
8. Use the Property of Equality for Exponential
Functions.
SOLUTION:
ANSWER:
9
Use the Property of Inequality for Exponential a + 2 3a + 1
Functions. 11. 81 = 3
SOLUTION:
ANSWER:
Solve each equation. Use the Property of Equality for Exponential
4x + 2 Functions.
9. 8 = 64
SOLUTION:
ANSWER:
−7
Use the Property of Equality for Exponential
Functions. b + 2 2 − 2b
12. 256 = 4
SOLUTION:
ANSWER:
0
x − 6 Use the Property of Equality for Exponential
10. 5 = 125 Functions.
SOLUTION:
Use the Property of Equality for Exponential
Functions. ANSWER:
−1
3c + 1 3c − 1
13. 9 = 27
SOLUTION:
ANSWER:
9
a + 2 3a + 1
11. 81 = 3
SOLUTION:
Use the Property of Equality for Exponential
Functions.
Use the Property of Equality for Exponential
Functions.
ANSWER:
ANSWER:
−7 2y + 4 y + 1
14. 8 = 16
b + 2 2 − 2b
12. 256 = 4 SOLUTION:
SOLUTION:
Use the Property of Equality for Exponential
Functions.
Use the Property of Equality for Exponential
Functions.
ANSWER:
−4
ANSWER:
−1
15. CCSS MODELINGIn 2009, My-Lien received
3c + 1 3c − 1 $10,000 from her grandmother. Her parents invested
13. 9 = 27 all of the money, and by 2021, the amount will have
grown to $16,960.
SOLUTION: a. Write an exponential function that could be used to
model the money y. Write the function in terms of x,
the number of years since 2009.
b. Assume that the amount of money continues to
grow at the same rate. What would be the balance in
the account in 2031?
Use the Property of Equality for Exponential
Functions. SOLUTION:
a.
Substitute 16780 for y 10000 for a and 12 for x in the
exponential function and simplify.
ANSWER:
2y + 4 y + 1 The exponential function that models the situation
14. 8 = 16
is .
SOLUTION:
b.
Substitute 22 for x in the modeled function and solve
for y.
Use the Property of Equality for Exponential
Functions.
ANSWER:
x
a
. y = 10,000(1.045)
b
. about $26,336.52
Write an exponential function for the graph that
ANSWER:
passes through the given points.
−4 16. (0, 6.4) and (3, 100)
SOLUTION:
15. CCSS MODELINGIn 2009, My-Lien received Substitute 100 for y and 6.4 for a and 3 for x into an
$10,000 from her grandmother. Her parents invested exponential function and determine the value of b.
all of the money, and by 2021, the amount will have
grown to $16,960.
a. Write an exponential function that could be used to
model the money y. Write the function in terms of x,
the number of years since 2009.
b. Assume that the amount of money continues to
grow at the same rate. What would be the balance in
the account in 2031?
SOLUTION:
a. An exponential function that passes through the given
Substitute 16780 for y 10000 for a and 12 for x in the points is .
exponential function and simplify.
ANSWER:
x
y = 6.4(2.5)
17. (0, 256) and (4, 81)
SOLUTION:
Substitute 81 for y and 256 for a and 4 for x into an
The exponential function that models the situation exponential function and determine the value of b.
is .
b.
Substitute 22 for x in the modeled function and solve
for y.
An exponential function that passes through the given
points is .
ANSWER:
x
a
. y = 10,000(1.045)
b ANSWER:
. about $26,336.52 x
y = 256(0.75)
Write an exponential function for the graph that 18. (0, 128) and (5, 371,293)
passes through the given points.
16. (0, 6.4) and (3, 100)
SOLUTION:
SOLUTION: Substitute 371293 for y and 128 for a and 5 for x into
Substitute 100 for y and 6.4 for a and 3 for x into an an exponential function and determine the value of b.
exponential function and determine the value of b.
An exponential function that passes through the given
An exponential function that passes through the given points is .
points is .
ANSWER:
x
ANSWER: y = 128(4.926)
x
y = 6.4(2.5)
19. (0, 144), and (4, 21,609)
17. (0, 256) and (4, 81) SOLUTION:
Substitute 21609 for y and 144 for a and 4 for x into
SOLUTION: an exponential function and determine the value of b.
Substitute 81 for y and 256 for a and 4 for x into an
exponential function and determine the value of b.
An exponential function that passes through the given
An exponential function that passes through the given points is .
points is .
ANSWER:
x
y = 144(3.5)
ANSWER:
x
y = 256(0.75) 20. Find the balance of an account after 7 years if $700
18. (0, 128) and (5, 371,293) is deposited into an account paying 4.3% interest
compounded monthly.
SOLUTION:
Substitute 371293 for y and 128 for a and 5 for x into SOLUTION:
an exponential function and determine the value of b. Use the compound interest formula.
Substitute $700 for P, 0.043 for r, 12 for n and 7 for t
and simplify.
An exponential function that passes through the given
points is . ANSWER:
$945.34
ANSWER: 21. Determine how much is in a retirement account after
x
y = 128(4.926) 20 years if $5000 was invested at 6.05% interest
compounded weekly.
19. (0, 144), and (4, 21,609)
SOLUTION:
SOLUTION: Use the compound interest formula.
Substitute 21609 for y and 144 for a and 4 for x into Substitute $5000 for P, 0.0605 for r, 52 for n and 20
an exponential function and determine the value of b. for t and simplify.
An exponential function that passes through the given ANSWER:
points is . $16,755.63
22. A savings account offers 0.7% interest compounded
ANSWER: bimonthly. If $110 is deposited in this account, what
x will the balance be after 15 years?
y = 144(3.5)
20. Find the balance of an account after 7 years if $700 SOLUTION:
is deposited into an account paying 4.3% interest Use the compound interest formula.
compounded monthly. Substitute $110 for P, 0.007 for r, 6 for n and 15 for t
and simplify.
SOLUTION:
Use the compound interest formula.
Substitute $700 for P, 0.043 for r, 12 for n and 7 for t
and simplify.
ANSWER:
$122.17
23. A college savings account pays 13.2% annual
interest compounded semiannually. What is the
ANSWER: balance of an account after 12 years if $21,000 was
$945.34 initially deposited?
21. Determine how much is in a retirement account after
20 years if $5000 was invested at 6.05% interest SOLUTION:
compounded weekly. Use the compound interest formula.
Substitute $21,000 for P, 0.132 for r, 2 for n and 12
SOLUTION: for t and simplify.
Use the compound interest formula.
Substitute $5000 for P, 0.0605 for r, 52 for n and 20
for t and simplify.
ANSWER:
$97,362.61
Solve each inequality.
ANSWER:
$16,755.63 24.
22. A savings account offers 0.7% interest compounded SOLUTION:
bimonthly. If $110 is deposited in this account, what
will the balance be after 15 years?
SOLUTION:
Use the compound interest formula. Use the Property of Inequality for Exponential
Substitute $110 for P, 0.007 for r, 6 for n and 15 for t Functions.
and simplify.
ANSWER:
25.
ANSWER:
SOLUTION:
$122.17
23. A college savings account pays 13.2% annual
interest compounded semiannually. What is the
balance of an account after 12 years if $21,000 was Use the Property of Inequality for Exponential
initially deposited? Functions.
SOLUTION:
Use the compound interest formula.
Substitute $21,000 for P, 0.132 for r, 2 for n and 12
for t and simplify.
ANSWER:
26.
ANSWER:
$97,362.61 SOLUTION:
Solve each inequality.
24.
SOLUTION:
Use the Property of Inequality for Exponential Use the Property of Inequality for Exponential
Functions. Functions.
ANSWER:
ANSWER:
25.
SOLUTION:
27.
SOLUTION:
Use the Property of Inequality for Exponential
Functions.
Use the Property of Inequality for Exponential
Functions.
ANSWER:
26.
ANSWER:
SOLUTION:
28.
SOLUTION:
Use the Property of Inequality for Exponential
Functions.
Use the Property of Inequality for Exponential
Functions.
ANSWER:
ANSWER:
27.
29.
SOLUTION:
SOLUTION:
Use the Property of Inequality for Exponential
Functions.
Use the Property of Inequality for Exponential
Functions.
ANSWER:
28.
ANSWER:
SOLUTION:
30. SCIENCEA mug of hot chocolate is 90°C at time t
= 0. It is surrounded by air at a constant temperature
of 20°C. If stirred steadily, its temperature in Celsius
−t
after t minutes will be y(t) = 20 + 70(1.071) .
a. Find the temperature of the hot chocolate after 15
minutes.
Use the Property of Inequality for Exponential b. Find the temperature of the hot chocolate after 30
Functions. minutes.
c.
The optimum drinking temperature is 60°C. Will
the mug of hot chocolate be at or below this
temperature after 10 minutes?
SOLUTION:
a.
Substitute 15 for t in the equation and simplify.
ANSWER:
b.
29.
Substitute 30 for t in the equation and simplify.
SOLUTION:
c.
Substitute 10 for t in the equation and simplify.
So, temperature of the hot chocolate will be below
60°C after 10 minutes.
Use the Property of Inequality for Exponential
Functions. ANSWER:
a
. 45.02° C
b
. 28.94° C
c. below
31. ANIMALSStudies show that an animal will defend
a territory, with area in square yards, that is directly
proportional to the 1.31 power of the animal’s weight
ANSWER: in pounds.
a. If a 45-pound beaver will defend 170 square yards,
write an equation for the area a defended by a
beaver weighing w pounds.
30. SCIENCEA mug of hot chocolate is 90°C at time t b. Scientists believe that thousands of years ago, the
= 0. It is surrounded by air at a constant temperature beaver’s ancestors were 11 feet long and weighed
of 20°C. If stirred steadily, its temperature in Celsius 430 pounds. Use your equation to determine the area
−t
after t minutes will be y(t) = 20 + 70(1.071) . defended by these animals.
a. Find the temperature of the hot chocolate after 15
minutes. SOLUTION:
b. Find the temperature of the hot chocolate after 30 a.
minutes. Substitute 170 for y, 45 for b, and 1.31 for x in the
exponential function.
c.
The optimum drinking temperature is 60°C. Will
the mug of hot chocolate be at or below this
temperature after 10 minutes?
SOLUTION:
a.
Substitute 15 for t in the equation and simplify.
The equation for the area a defended by a beaver
weighting w pounds is
b. b.
Substitute 30 for t in the equation and simplify. Substitute 430 for w in the equation and solve for y.
c.
Substitute 10 for t in the equation and simplify.
ANSWER:
a 1.31
. a = 1.16w
b 2
So, temperature of the hot chocolate will be below . about 3268 yd
60°C after 10 minutes. Solve each equation.
ANSWER:
a 32.
. 45.02° C
b
. 28.94° C
c. below SOLUTION:
31. ANIMALSStudies show that an animal will defend
a territory, with area in square yards, that is directly
proportional to the 1.31 power of the animal’s weight
in pounds.
a. If a 45-pound beaver will defend 170 square yards,
write an equation for the area a defended by a Use the Property of Equality for Exponential
beaver weighing w pounds.
b. Scientists believe that thousands of years ago, the Functions.
beaver’s ancestors were 11 feet long and weighed
430 pounds. Use your equation to determine the area
defended by these animals.
SOLUTION:
a.
Substitute 170 for y, 45 for b, and 1.31 for x in the
exponential function. ANSWER:
33.
The equation for the area a defended by a beaver
weighting w pounds is SOLUTION:
b.
Substitute 430 for w in the equation and solve for y.
Use the Property of Equality for Exponential
ANSWER: Functions.
a 1.31
. a = 1.16w
b 2
. about 3268 yd
Solve each equation.
32.
ANSWER:
SOLUTION:
34.
SOLUTION:
Use the Property of Equality for Exponential
Functions.
Use the Property of Equality for Exponential
Functions.
ANSWER:
ANSWER:
−6
33.
35.
SOLUTION:
SOLUTION:
Use the Property of Equality for Exponential
Functions.
Use the Property of Equality for Exponential
Functions.
ANSWER:
ANSWER:
34.
SOLUTION:
36.
SOLUTION:
Use the Property of Equality for Exponential
Functions.
ANSWER: Use the Property of Equality for Exponential
−6
Functions.
35.
SOLUTION:
ANSWER:
37.
Use the Property of Equality for Exponential
Functions. SOLUTION:
Use the Property of Equality for Exponential
ANSWER: Functions.
36.
ANSWER:
SOLUTION:
1
38. CCSS MODELINGIn 1950, the world population
was about 2.556 billion. By 1980, it had increased to
about 4.458 billion.
a. x
Write an exponential function of the form y = ab
that could be used to model the world population y in
billions for 1950 to 1980. Write the equation in terms
of x, the number of years since 1950. (Round the
Use the Property of Equality for Exponential value of b to the nearest ten-thousandth.)
Functions. b. Suppose the population continued to grow at that
rate. Estimate the population in 2000.
c. In 2000, the population of the world was about
6.08 billion. Compare your estimate to the actual
population.
d. Use the equation you wrote in Part a to estimate
the world population in the year 2020. How accurate
do you think the estimate is? Explain your reasoning.
ANSWER:
SOLUTION:
a.
Substitute 4.458 for y, 2.556 for a, and 30 for x in the
exponential function and solve for b.
37.
SOLUTION:
The exponential function that model the situation is
.
b.
Use the Property of Equality for Exponential Substitute 50 for x in the equation and simplify.
Functions.
c. The prediction was about 375 million greater than
the actual population.
d.
ANSWER:
1 Substitute 70 for x in the equation and simplify.
38. CCSS MODELINGIn 1950, the world population
was about 2.556 billion. By 1980, it had increased to
about 4.458 billion.
a. x
Write an exponential function of the form y = ab Because the prediction for 2000 was greater than the
that could be used to model the world population y in actual population, this prediction for 2020 is probably
billions for 1950 to 1980. Write the equation in terms even higher than the actual population will be at the
of x, the number of years since 1950. (Round the time.
value of b to the nearest ten-thousandth.)
b. Suppose the population continued to grow at that
rate. Estimate the population in 2000. ANSWER:
x
a
. y = 2.556(1.0187)
c. In 2000, the population of the world was about b
6.08 billion. Compare your estimate to the actual . 6.455 billion
population. c. The prediction was about 375 million greater than
d. Use the equation you wrote in Part a to estimate the actual.
d
the world population in the year 2020. How accurate . About 9.3498 billion; because the prediction for
do you think the estimate is? Explain your reasoning. 2000 was greater than the actual population, this
prediction is probably even higher than the actual
SOLUTION: population will be at the time.
a.
Substitute 4.458 for y, 2.556 for a, and 30 for x in the 39. TREES The diameter of the base of a tree trunk in
exponential function and solve for b. centimeters varies directly with the power of its
height in meters.
a. A young sequoia tree is 6 meters tall, and the
diameter of its base is 19.1 centimeters. Use this
information to write an equation for the diameter d of
The exponential function that model the situation is the base of a sequoia tree if its height is h meters
high
. b. The General Sherman Tree in Sequoia National
b. Park, California, is approximately 84 meters tall.
Substitute 50 for x in the equation and simplify. Find the diameter of the General Sherman Tree at its
base.
SOLUTION:
a.
The equation that represent the situation is
c. The prediction was about 375 million greater than
the actual population. .
d. b.
Substitute 70 for x in the equation and simplify. Substitute 84 for h in the equation and solve for d.
Because the prediction for 2000 was greater than the The diameter of the General Sherman Tree at its
actual population, this prediction for 2020 is probably base is about 1001 cm.
even higher than the actual population will be at the
time.
ANSWER:
ANSWER: a.
x
a
. y = 2.556(1.0187) b
. about 1001 cm
b
. 6.455 billion
c. The prediction was about 375 million greater than 40. FINANCIAL LITERACYMrs. Jackson has two
the actual. different retirement investment plans from which to
d choose.
. About 9.3498 billion; because the prediction for
2000 was greater than the actual population, this a. Write equations for Option A and Option B given
prediction is probably even higher than the actual the minimum deposits.
population will be at the time. b. Draw a graph to show the balances for each
39. TREES The diameter of the base of a tree trunk in investment option after t years.
c. Explain whether Option A or Option B is the
centimeters varies directly with the power of its better investment choice.
height in meters.
a. A young sequoia tree is 6 meters tall, and the
diameter of its base is 19.1 centimeters. Use this
information to write an equation for the diameter d of
the base of a sequoia tree if its height is h meters
high
b. The General Sherman Tree in Sequoia National
Park, California, is approximately 84 meters tall.
Find the diameter of the General Sherman Tree at its
SOLUTION:
base.
a.
Use the compound interest formula.
SOLUTION: The equation that represents Option A
a. is .
The equation that represent the situation is
. The equation that represents Option B
b.
Substitute 84 for h in the equation and solve for d. is
b.
The graph that shows the balances for each
investment option after t years:
The diameter of the General Sherman Tree at its
base is about 1001 cm.
ANSWER:
a.
b
. about 1001 cm
40. FINANCIAL LITERACYMrs. Jackson has two
different retirement investment plans from which to
choose.
a. Write equations for Option A and Option B given
the minimum deposits.
b. Draw a graph to show the balances for each
c.
investment option after t years. During the first 22 years, Option B is the better
c. Explain whether Option A or Option B is the choice because the total is greater than that of
better investment choice. Option A. However, after about 22 years, the
balance of Option A exceeds that of Option B, so
Option A is the better choice.
ANSWER:
a.
b.
SOLUTION:
a.
Use the compound interest formula.
The equation that represents Option A
is .
The equation that represents Option B
is
b.
Sample answer:
The graph that shows the balances for each c. During the first 22 years, Option
investment option after t years: B is the better choice because the total is greater
than that of Option A. However, after about 22
years, the balance of Option A exceeds that of
Option B, so Option A is the better choice.
41. MULTIPLE REPRESENTATIONSIn this
problem, you will explore the rapid increase of an
exponential function. A large sheet of paper is cut in
half, and one of the resulting pieces is placed on top
of the other. Then the pieces in the stack are cut in
half and placed on top of each other. Suppose this
procedure is repeated several times.
a. CONCRETE
Perform this activity and count the
number of sheets in the stack after the first cut. How
many pieces will there be after the second cut? How
many pieces after the third cut? How many pieces
c. after the fourth cut?
During the first 22 years, Option B is the better b. TABULAR
choice because the total is greater than that of Record your results in a table.
c. SYMBOLIC
Option A. However, after about 22 years, the Use the pattern in the table to write
balance of Option A exceeds that of Option B, so an equation for the number of pieces in the stack
Option A is the better choice. after x cuts.
d. ANALYTICAL
The thickness of ordinary paper
is about 0.003 inch. Write an equation for the
ANSWER:
a. thickness of the stack of paper after x cuts.
e.ANALYTICAL
How thick will the stack of
b. paper be after 30 cuts?
SOLUTION:
a.
There will be 2, 4, 8, 16 pieces after the first, second,
third and fourth cut respectively.
b.
c.
The equation that represent the situation is
d.
Sample answer:
c. During the first 22 years, Option Substitute 0.003 for a and 2 for b in the exponential
B is the better choice because the total is greater function.
than that of Option A. However, after about 22
years, the balance of Option A exceeds that of
Option B, so Option A is the better choice.
e.
41. MULTIPLE REPRESENTATIONSIn this Substitute 30 for x in the equation and
problem, you will explore the rapid increase of an simplify.
exponential function. A large sheet of paper is cut in
half, and one of the resulting pieces is placed on top
of the other. Then the pieces in the stack are cut in
half and placed on top of each other. Suppose this
procedure is repeated several times.
a. CONCRETE The thickness of the stack of paper after 30 cuts is
Perform this activity and count the about 3221225.47 in.
number of sheets in the stack after the first cut. How
many pieces will there be after the second cut? How
many pieces after the third cut? How many pieces ANSWER:
after the fourth cut? a. 2, 4, 8, 16
b
b. TABULAR .
Record your results in a table.
c. SYMBOLIC
Use the pattern in the table to write
an equation for the number of pieces in the stack
after x cuts.
d. ANALYTICAL
The thickness of ordinary paper
is about 0.003 inch. Write an equation for the
thickness of the stack of paper after x cuts. x
c
e.ANALYTICAL . y = 2
How thick will the stack of
x
paper be after 30 cuts? d
. y = 0.003(2)
e. about 3,221,225.47 in.
SOLUTION:
a.
There will be 2, 4, 8, 16 pieces after the first, second, 42. WRITING IN MATHIn a problem about
third and fourth cut respectively. compound interest, describe what happens as the
b. compounding period becomes more frequent while
the principal and overall time remain the same.
SOLUTION:
Sample answer: The more frequently interest is
compounded, the higher the account balance
becomes.
ANSWER:
c.
The equation that represent the situation is Sample answer: The more frequently interest is
d. compounded, the higher the account balance
Substitute 0.003 for a and 2 for b in the exponential becomes.
function. x −
ERROR ANALYSIS
43. Beth and Liz are solving 6
3 > 36−x − 1. Is either of them correct? Explain your
reasoning.
e.
Substitute 30 for x in the equation and
simplify.
The thickness of the stack of paper after 30 cuts is
about 3221225.47 in.
ANSWER:
a. 2, 4, 8, 16
b
.
x
c. y = 2
x
d
. y = 0.003(2)
e. about 3,221,225.47 in.
42. WRITING IN MATHIn a problem about
compound interest, describe what happens as the
compounding period becomes more frequent while SOLUTION:
the principal and overall time remain the same. Sample answer: Beth; Liz added the exponents
instead of multiplying them when taking the power of
SOLUTION:
a power.
Sample answer: The more frequently interest is
compounded, the higher the account balance ANSWER:
becomes. Sample answer: Beth; Liz added the exponents
instead of multiplying them when taking the power of
ANSWER:
a power.
Sample answer: The more frequently interest is
compounded, the higher the account balance 18 18 18
becomes. 44. CHALLENGESolve for x: 16 + 16 + 16 +
18 18 x
16 + 16 = 4 .
x −
ERROR ANALYSIS
43. Beth and Liz are solving 6
SOLUTION:
3 > 36−x − 1. Is either of them correct? Explain your
reasoning.
ANSWER:
37.1610
45. OPEN ENDEDWhat would be a more beneficial
change to a 5-year loan at 8% interest compounded
monthly: reducing the term to 4 years or reducing the
interest rate to 6.5%?
SOLUTION:
Reducing the term will be more beneficial. The
multiplier is 1.3756 for the 4-year and 1.3828 for the
6.5%.
ANSWER:
Reducing the term will be more beneficial. The
multiplier is 1.3756 for the 4-year and 1.3828 for the
6.5%.
SOLUTION: CCSS ARGUMENTS
46. Determine whether the
Sample answer: Beth; Liz added the exponents following statements are sometimes, always, or
instead of multiplying them when taking the power of never true. Explain your reasoning.
x 20x
a power. a. 2 > −8 for all values of x.
b. The graph of an exponential growth equation is
ANSWER: increasing.
Sample answer: Beth; Liz added the exponents
instead of multiplying them when taking the power of c. The graph of an exponential decay equation is
increasing.
a power.
18 18 18 SOLUTION:
44. CHALLENGESolve for x: 16 + 16 + 16 + a. x 20x
18 18 x Always; 2 will always be positive, and −8 will
16 + 16 = 4 . always be negative.
b. Always; by definition the graph will always be
SOLUTION: increasing even if it is a small increase.
c. Never; by definition the graph will always be
decreasing even if it is a small decrease.
ANSWER:
a x 20x
. Always; 2 will always be positive, and 8 will
−
always be negative.
b
. Always; by definition the graph will always be
increasing even if it is a small increase.
c. Never; by definition the graph will always be
decreasing even if it is a small decrease.
ANSWER:
37.1610
OPEN ENDEDWrite an exponential inequality with
47.
a solution of x 2.
≤
45. OPEN ENDEDWhat would be a more beneficial
change to a 5-year loan at 8% interest compounded SOLUTION:
monthly: reducing the term to 4 years or reducing the x 2
Sample answer: 4 4
interest rate to 6.5%? ≤
ANSWER:
SOLUTION: x 2
Sample answer: 4 4
Reducing the term will be more beneficial. The ≤
multiplier is 1.3756 for the 4-year and 1.3828 for the 2x x + 1 2x + 2 4x + 1
6.5%. PROOFShow that 27 · 81 = 3 · 9 .
48.
SOLUTION:
ANSWER:
Reducing the term will be more beneficial. The
multiplier is 1.3756 for the 4-year and 1.3828 for the
6.5%.
CCSS ARGUMENTS
46. Determine whether the
following statements are sometimes, always, or
never true. Explain your reasoning.
a. x 20x
2 > −8 for all values of x.
b. The graph of an exponential growth equation is
increasing.
c. The graph of an exponential decay equation is
increasing.
ANSWER:
SOLUTION:
a. x 20x
Always; 2 will always be positive, and −8 will
always be negative.
b. Always; by definition the graph will always be
increasing even if it is a small increase.
c. Never; by definition the graph will always be
decreasing even if it is a small decrease.
WRITING IN MATHIf you were given the initial
49.
ANSWER: and final amounts of a radioactive substance and the
a x 20x
. Always; 2 will always be positive, and 8 will
− amount of time that passes, how would you
always be negative. determine the rate at which the amount was
b
. Always; by definition the graph will always be increasing or decreasing in order to write an
increasing even if it is a small increase. equation?
c. Never; by definition the graph will always be
decreasing even if it is a small decrease. SOLUTION:
Sample answer: Divide the final amount by the initial
OPEN ENDEDWrite an exponential inequality with amount. If n is the number of time intervals that pass,
47.
a solution of x 2. take the nth root of the answer.
≤
SOLUTION: ANSWER:
x 2 Sample answer: Divide the final amount by the initial
Sample answer: 4 4
≤ amount. If n is the number of time intervals that pass,
take the nth root of the answer.
ANSWER:
x 2
Sample answer: 4 4
≤ −4 =
50. 3 × 10
A 30,000
2x x + 1 2x + 2 4x + 1 −
PROOFShow that 27 · 81 = 3 · 9 .
48. B 0.0003
SOLUTION: C 120
−
D 0.00003
SOLUTION:
B is the correct option.
ANSWER:
ANSWER:
B
51. Which of the following could not be a solution to 5 −
3x < 3?
−
F2.5
G3
H 3.5
J
4
WRITING IN MATHIf you were given the initial
49.
and final amounts of a radioactive substance and the SOLUTION:
amount of time that passes, how would you Check the inequality by substituting 2.5 for x.
determine the rate at which the amount was
increasing or decreasing in order to write an
equation?
SOLUTION: So, F is the correct option.
Sample answer: Divide the final amount by the initial
amount. If n is the number of time intervals that pass,
take the nth root of the answer. ANSWER:
F
ANSWER:
GRIDDED RESPONSEThe three angles of a
Sample answer: Divide the final amount by the initial 52.
amount. If n is the number of time intervals that pass, triangle are 3x, x + 10, and 2x − 40. Find the measure
take the nth root of the answer. of the smallest angle in the triangle.
−4 SOLUTION:
= Sum of the three angles in a triangle is 180 .
50. 3 × 10 º
A 30,000
−
B 0.0003
C 120
−
D 0.00003
SOLUTION:
The measure of the smallest angle in the triangle is
B is the correct option. 30 .
º
ANSWER:
B ANSWER:
30
51. Which of the following could not be a solution to 5 − SAT/ACT Which of the following is equivalent to
3x < 3? 53.
− (x)(x)(x)(x) for all x?
F2.5
G3 A x + 4
H 3.5 B 4x
J
4 C 2x2
2
SOLUTION: D 4x
Check the inequality by substituting 2.5 for x. E 4
x
SOLUTION:
So, F is the correct option.
E is the correct choice.
ANSWER:
F
ANSWER:
E
GRIDDED RESPONSEThe three angles of a
52.
triangle are 3x, x + 10, and 2x − 40. Find the measure Graph each function.
of the smallest angle in the triangle. x
y = 2(3)
54.
SOLUTION:
Sum of the three angles in a triangle is 180 . SOLUTION:
º Make a table of values. Then plot the points and
sketch the graph.
The measure of the smallest angle in the triangle is
30 .
º
ANSWER:
30
SAT/ACT Which of the following is equivalent to
53.
(x)(x)(x)(x) for all x?
A x + 4
B 4x
2 ANSWER:
C2x
D 4x2
E 4
x
SOLUTION:
E is the correct choice.
ANSWER:
x
E
y = 5(2)
55.
Graph each function. SOLUTION:
x
y = 2(3) Make a table of values. Then plot the points and
54. sketch the graph.
SOLUTION:
Make a table of values. Then plot the points and
sketch the graph.
ANSWER:
ANSWER:
56.
x
y = 5(2) SOLUTION:
55. Make a table of values. Then plot the points and
SOLUTION: sketch the graph.
Make a table of values. Then plot the points and
sketch the graph.
ANSWER:
ANSWER:
Solve each equation.
57.
SOLUTION:
56.
SOLUTION:
Make a table of values. Then plot the points and
sketch the graph.
ANSWER:
4
58.
SOLUTION:
ANSWER:
18
59.
ANSWER: SOLUTION:
ANSWER:
8.5
Solve each equation.
60.
57.
SOLUTION:
SOLUTION:
ANSWER: The square root of x cannot be negative, so there is
4
no solution.
ANSWER:
58. no solution
SOLUTION:
61.
SOLUTION:
ANSWER:
18
ANSWER:
59. 5
SOLUTION:
62.
SOLUTION:
ANSWER:
8.5
60.
ANSWER:
SOLUTION: 20
−
63.
SOLUTION:
The square root of x cannot be negative, so there is
no solution.
ANSWER:
no solution
ANSWER:
61. 5
SOLUTION:
64.
SOLUTION:
ANSWER:
5
62.
SOLUTION:
ANSWER:
65.
SOLUTION:
ANSWER:
20
−
63.
SOLUTION:
ANSWER:
1
−
SALES A salesperson earns $10 an hour plus a 10%
66.
commission on sales. Write a function to describe the
salesperson’s income. If the salesperson wants to
earn $1000 in a 40-hour week, what should his sales
be?
ANSWER: SOLUTION:
5 Let I be the income of the salesperson and m be his
sales.
The function that represent the situation is
64. .
SOLUTION: Substitute 1000 for I in the equation and solve for m.
ANSWER:
I(m) = 400 + 0.1m; $6000
STATE FAIRA dairy makes three types of
67.
cheese cheddar, Monterey Jack, and Swiss and
— —
sells the cheese in three booths at the state fair. At
ANSWER: the beginning of one day, the first booth received x
pounds of each type of cheese. The second booth
received y pounds of each type of cheese, and the
third booth received z pounds of each type of cheese.
By the end of the day, the dairy had sold 131 pounds
65. of cheddar, 291 pounds of Monterey Jack, and 232
pounds of Swiss. The table below shows the percent
SOLUTION: of the cheese delivered in the morning that was sold
at each booth. How many pounds of cheddar cheese
did each booth receive in the morning?
ANSWER:
SOLUTION:
1
− The system of equations that represent the situation:
SALES A salesperson earns $10 an hour plus a 10%
66.
commission on sales. Write a function to describe the
salesperson’s income. If the salesperson wants to
earn $1000 in a 40-hour week, what should his sales
be?
SOLUTION: Eliminate the variable x by using two pairs of
Let I be the income of the salesperson and m be his equations.
sales.
The function that represent the situation is Subtract (1) and (2).
.
Substitute 1000 for I in the equation and solve for m.
Multiply (2) by 3 and (3) by 4 and subtract both the
ANSWER: equations.
I(m) = 400 + 0.1m; $6000
STATE FAIRA dairy makes three types of
67.
cheese cheddar, Monterey Jack, and Swiss and
— —
sells the cheese in three booths at the state fair. At
the beginning of one day, the first booth received x
pounds of each type of cheese. The second booth
received y pounds of each type of cheese, and the Solve the system of two equations:
third booth received z pounds of each type of cheese.
By the end of the day, the dairy had sold 131 pounds
of cheddar, 291 pounds of Monterey Jack, and 232
pounds of Swiss. The table below shows the percent
of the cheese delivered in the morning that was sold
at each booth. How many pounds of cheddar cheese
did each booth receive in the morning?
Substitute z = 100 in the equation
SOLUTION:
The system of equations that represent the situation:
Substitute y = 150 and z = 100 in the (1) and solve
for x.
Eliminate the variable x by using two pairs of
equations.
Booth 1 has 190 lb; Booth 2 has 150 lb; Booth 3 has
Subtract (1) and (2). 100 lb.
ANSWER:
booth 1, 190 lb; booth 2, 150 lb; booth 3, 100 lb
Find [g h](x) and [h g](x).
◦ ◦
h(x) = 2x 1
Multiply (2) by 3 and (3) by 4 and subtract both the 68. −
equations. g(x) = 3x + 4
SOLUTION:
Solve the system of two equations:
ANSWER:
6x + 1; 6x + 7
h(x) = x2 + 2
69.
Substitute z = 100 in the equation g(x) = x − 3
SOLUTION:
Substitute y = 150 and z = 100 in the (1) and solve
for x.
ANSWER:
2 2
x 1; x 6x + 11
− −
h(x) = x2 + 1
70.
g(x) = 2x + 1
−
SOLUTION:
Booth 1 has 190 lb; Booth 2 has 150 lb; Booth 3 has
100 lb.
ANSWER:
booth 1, 190 lb; booth 2, 150 lb; booth 3, 100 lb
Find [g h](x) and [h g](x).
◦ ◦
68. h(x) = 2x − 1
g(x) = 3x + 4 ANSWER:
2x2 1; 4x2 4x + 2
− − −
SOLUTION:
h(x) = 5x
71. −
g(x) = 3x − 5
SOLUTION:
ANSWER:
6x + 1; 6x + 7
ANSWER:
15x 5; 15x + 25
2 − − −
69. h(x) = x + 2 3
g(x) = x − 3 h(x) = x
72.
g(x) = x − 2
SOLUTION:
SOLUTION:
ANSWER:
3 3 2
x x 6x + 12x 8
−2; − −
ANSWER:
x2 1; x2 6x + 11
− − h(x) = x + 4
73.
2 g(x) = | x |
70. h(x) = x + 1
g(x) = 2x + 1 SOLUTION:
−
SOLUTION:
ANSWER:
| x + 4 | ; | x | + 4
ANSWER:
2x2 1; 4x2 4x + 2
− − −
h(x) = 5x
71. −
g(x) = 3x − 5
SOLUTION:
ANSWER:
15x 5; 15x + 25
− − −
h(x) = x3
72.
g(x) = x − 2
SOLUTION:
ANSWER:
3 3 2
x x 6x + 12x 8
−2; − −
73. h(x) = x + 4
g(x) = | x |
SOLUTION:
ANSWER:
| x + 4 | ; | x | + 4
Solve each equation.
5x 2x − 4
1. 3 = 27
SOLUTION:
Use the Property of Equality for Exponential
Functions.
ANSWER:
12
2y − 3 y + 1
2. 16 = 4
SOLUTION:
Use the Property of Equality for Exponential
Functions.
ANSWER:
6x x − 2
3. 2 = 32
SOLUTION:
Use the Property of Equality for Exponential
Functions.
Solve each equation.
5x 2x − 4
1. 3 = 27
SOLUTION:
ANSWER:
−10
x + 5 8x − 6
4. 49 = 7
Use the Property of Equality for Exponential SOLUTION:
Functions.
ANSWER: Use the Property of Equality for Exponential
12 Functions.
2y − 3 y + 1
2. 16 = 4
SOLUTION:
ANSWER:
Use the Property of Equality for Exponential
Functions. SCIENCE
5. Mitosis is a process in which one cell
divides into two. The Escherichia coli is one of the
fastest growing bacteria. It can reproduce itself in 15
minutes.
a. Write an exponential function to represent the
number of cells c after t minutes.
b. If you begin with one Escherichia coli cell, how
many cells will there be in one hour?
ANSWER:
SOLUTION:
a.
The exponential function that represent the number
6x x − 2 of cells after t minutes is .
3. 2 = 32
b.
SOLUTION: Substitute 1 for t in the function and solve for c.
ANSWER:
Use the Property of Equality for Exponential a.
Functions. b. 16 cells
6. A certificate of deposit (CD) pays 2.25% annual
interest compounded biweekly. If you deposit $500
into this CD, what will the balance be after 6 years?
ANSWER:
−10 SOLUTION:
Use the compound interest formula.
x + 5 8x − 6 Substitute $500 for P, 0.0225 for r, 26 for n and 6 for
4. 49 = 7 t and simplify.
SOLUTION:
Use the Property of Equality for Exponential
Functions.
ANSWER:
$572.23
Solve each inequality.
2x + 6 2x – 4
7. 4 ≤64
SOLUTION:
ANSWER:
SCIENCE
5. Mitosis is a process in which one cell Use the Property of Inequality for Exponential
divides into two. The Escherichia coli is one of the Functions.
fastest growing bacteria. It can reproduce itself in 15
minutes.
a. Write an exponential function to represent the
number of cells c after t minutes.
b. If you begin with one Escherichia coli cell, how
many cells will there be in one hour?
ANSWER:
SOLUTION:
a. x ≥ 4.5
The exponential function that represent the number
of cells after t minutes is .
b. 8.
Substitute 1 for t in the function and solve for c.
SOLUTION:
ANSWER:
a.
b. 16 cells
Use the Property of Inequality for Exponential
6. A certificate of deposit (CD) pays 2.25% annual Functions.
interest compounded biweekly. If you deposit $500
into this CD, what will the balance be after 6 years?
SOLUTION:
Use the compound interest formula.
Substitute $500 for P, 0.0225 for r, 26 for n and 6 for ANSWER:
t and simplify.
Solve each equation.
4x + 2
9. 8 = 64
SOLUTION:
Use the Property of Equality for Exponential
ANSWER: Functions.
$572.23
Solve each inequality.
2x + 6 2x – 4
7. 4 ≤64
ANSWER:
SOLUTION:
0
x − 6
10. 5 = 125
SOLUTION:
Use the Property of Inequality for Exponential
Functions.
Use the Property of Equality for Exponential
Functions.
ANSWER:
x ≥ 4.5
ANSWER:
9
8. a + 2 3a + 1
11. 81 = 3
SOLUTION:
SOLUTION:
Use the Property of Inequality for Exponential Use the Property of Equality for Exponential
Functions. Functions.
ANSWER:
ANSWER:
7-2 Solving Exponential Equations and Inequalities
−7
Solve each equation. b + 2 2 − 2b
4x + 2 12. 256 = 4
9. 8 = 64
SOLUTION:
SOLUTION:
Use the Property of Equality for Exponential Use the Property of Equality for Exponential
Functions.
Functions.
ANSWER:
0
ANSWER:
−1
x − 6
10. 5 = 125
3c + 1 3c − 1
13. 9 = 27
SOLUTION:
SOLUTION:
Use the Property of Equality for Exponential
Functions.
Use the Property of Equality for Exponential
Functions.
ANSWER:
9
a + 2 3a + 1
11. 81 = 3
SOLUTION:
ANSWER:
2y + 4 y + 1
14. 8 = 16
Use the Property of Equality for Exponential
Functions. SOLUTION:
ANSWER:
−7 Use the Property of Equality for Exponential
Functions.
b + 2 2 − 2b
12. 256 = 4
eSolutions Manual - Powered by Cognero Page3
SOLUTION:
ANSWER:
−4
Use the Property of Equality for Exponential
Functions. 15. CCSS MODELINGIn 2009, My-Lien received
$10,000 from her grandmother. Her parents invested
all of the money, and by 2021, the amount will have
grown to $16,960.
a. Write an exponential function that could be used to
model the money y. Write the function in terms of x,
the number of years since 2009.
b. Assume that the amount of money continues to
ANSWER:
−1 grow at the same rate. What would be the balance in
the account in 2031?
3c + 1 3c − 1
13. 9 = 27
SOLUTION:
a.
SOLUTION: Substitute 16780 for y 10000 for a and 12 for x in the
exponential function and simplify.
Use the Property of Equality for Exponential
Functions.
The exponential function that models the situation
is .
b.
ANSWER: Substitute 22 for x in the modeled function and solve
for y.
2y + 4 y + 1
14. 8 = 16
SOLUTION:
ANSWER:
x
a
. y = 10,000(1.045)
b
. about $26,336.52
Write an exponential function for the graph that
Use the Property of Equality for Exponential passes through the given points.
Functions. 16. (0, 6.4) and (3, 100)
SOLUTION:
Substitute 100 for y and 6.4 for a and 3 for x into an
exponential function and determine the value of b.
ANSWER:
−4
15. CCSS MODELINGIn 2009, My-Lien received
$10,000 from her grandmother. Her parents invested
all of the money, and by 2021, the amount will have
grown to $16,960. An exponential function that passes through the given
a. Write an exponential function that could be used to points is .
model the money y. Write the function in terms of x,
the number of years since 2009.
b. Assume that the amount of money continues to ANSWER:
x
grow at the same rate. What would be the balance in y = 6.4(2.5)
the account in 2031?
17. (0, 256) and (4, 81)
SOLUTION:
a. SOLUTION:
Substitute 16780 for y 10000 for a and 12 for x in the Substitute 81 for y and 256 for a and 4 for x into an
exponential function and simplify. exponential function and determine the value of b.
The exponential function that models the situation
is . An exponential function that passes through the given
points is .
b.
Substitute 22 for x in the modeled function and solve ANSWER:
for y. x
y = 256(0.75)
18. (0, 128) and (5, 371,293)
SOLUTION:
Substitute 371293 for y and 128 for a and 5 for x into
ANSWER: an exponential function and determine the value of b.
x
a
. y = 10,000(1.045)
b
. about $26,336.52
Write an exponential function for the graph that
passes through the given points.
16. (0, 6.4) and (3, 100)
SOLUTION:
Substitute 100 for y and 6.4 for a and 3 for x into an
exponential function and determine the value of b. An exponential function that passes through the given
points is .
ANSWER:
x
y = 128(4.926)
19. (0, 144), and (4, 21,609)
SOLUTION:
An exponential function that passes through the given Substitute 21609 for y and 144 for a and 4 for x into
points is . an exponential function and determine the value of b.
ANSWER:
x
y = 6.4(2.5)
17. (0, 256) and (4, 81)
SOLUTION:
Substitute 81 for y and 256 for a and 4 for x into an An exponential function that passes through the given
exponential function and determine the value of b.
points is .
ANSWER:
x
y = 144(3.5)
20. Find the balance of an account after 7 years if $700
is deposited into an account paying 4.3% interest
compounded monthly.
An exponential function that passes through the given SOLUTION:
points is . Use the compound interest formula.
Substitute $700 for P, 0.043 for r, 12 for n and 7 for t
and simplify.
ANSWER:
x
y = 256(0.75)
18. (0, 128) and (5, 371,293)
SOLUTION:
Substitute 371293 for y and 128 for a and 5 for x into
an exponential function and determine the value of b.
ANSWER:
$945.34
21. Determine how much is in a retirement account after
20 years if $5000 was invested at 6.05% interest
compounded weekly.
SOLUTION:
An exponential function that passes through the given Use the compound interest formula.
points is . Substitute $5000 for P, 0.0605 for r, 52 for n and 20
for t and simplify.
ANSWER:
x
y = 128(4.926)
19. (0, 144), and (4, 21,609)
SOLUTION:
Substitute 21609 for y and 144 for a and 4 for x into
an exponential function and determine the value of b.
ANSWER:
$16,755.63
22. A savings account offers 0.7% interest compounded
bimonthly. If $110 is deposited in this account, what
will the balance be after 15 years?
SOLUTION:
Use the compound interest formula.
An exponential function that passes through the given Substitute $110 for P, 0.007 for r, 6 for n and 15 for t
points is . and simplify.
ANSWER:
x
y = 144(3.5)
20. Find the balance of an account after 7 years if $700
is deposited into an account paying 4.3% interest
compounded monthly.
ANSWER:
SOLUTION:
Use the compound interest formula. $122.17
Substitute $700 for P, 0.043 for r, 12 for n and 7 for t 23. A college savings account pays 13.2% annual
and simplify. interest compounded semiannually. What is the
balance of an account after 12 years if $21,000 was
initially deposited?
SOLUTION:
Use the compound interest formula.
Substitute $21,000 for P, 0.132 for r, 2 for n and 12
for t and simplify.
ANSWER:
$945.34
21. Determine how much is in a retirement account after
20 years if $5000 was invested at 6.05% interest
compounded weekly.
SOLUTION:
Use the compound interest formula. ANSWER:
Substitute $5000 for P, 0.0605 for r, 52 for n and 20 $97,362.61
for t and simplify. Solve each inequality.
24.
SOLUTION:
Use the Property of Inequality for Exponential
Functions.
ANSWER:
$16,755.63
22. A savings account offers 0.7% interest compounded
bimonthly. If $110 is deposited in this account, what
will the balance be after 15 years?
ANSWER:
SOLUTION:
Use the compound interest formula.
Substitute $110 for P, 0.007 for r, 6 for n and 15 for t
and simplify. 25.
SOLUTION:
Use the Property of Inequality for Exponential
Functions.
ANSWER:
$122.17
23. A college savings account pays 13.2% annual
interest compounded semiannually. What is the
balance of an account after 12 years if $21,000 was ANSWER:
initially deposited?
SOLUTION:
Use the compound interest formula.
Substitute $21,000 for P, 0.132 for r, 2 for n and 12 26.
for t and simplify.
SOLUTION:
ANSWER:
$97,362.61 Use the Property of Inequality for Exponential
Solve each inequality. Functions.
24.
SOLUTION:
Use the Property of Inequality for Exponential ANSWER:
Functions.
27.
SOLUTION:
ANSWER:
25.
SOLUTION:
Use the Property of Inequality for Exponential
Functions.
Use the Property of Inequality for Exponential
Functions.
ANSWER:
ANSWER: 28.
SOLUTION:
26.
SOLUTION:
Use the Property of Inequality for Exponential
Functions.
Use the Property of Inequality for Exponential
Functions.
ANSWER:
29.
ANSWER:
SOLUTION:
27.
SOLUTION:
Use the Property of Inequality for Exponential
Functions.
Use the Property of Inequality for Exponential
Functions.
ANSWER:
30. SCIENCEA mug of hot chocolate is 90°C at time t
ANSWER: = 0. It is surrounded by air at a constant temperature
of 20°C. If stirred steadily, its temperature in Celsius
−t
after t minutes will be y(t) = 20 + 70(1.071) .
28. a. Find the temperature of the hot chocolate after 15
minutes.
SOLUTION: b. Find the temperature of the hot chocolate after 30
minutes.
c.
The optimum drinking temperature is 60°C. Will
the mug of hot chocolate be at or below this
temperature after 10 minutes?
SOLUTION:
a.
Substitute 15 for t in the equation and simplify.
Use the Property of Inequality for Exponential
Functions.
b.
Substitute 30 for t in the equation and simplify.
c.
ANSWER: Substitute 10 for t in the equation and simplify.
29. So, temperature of the hot chocolate will be below
60°C after 10 minutes.
SOLUTION:
ANSWER:
a
. 45.02° C
b
. 28.94° C
c. below
31. ANIMALSStudies show that an animal will defend
a territory, with area in square yards, that is directly
Use the Property of Inequality for Exponential proportional to the 1.31 power of the animal’s weight
in pounds.
Functions. a. If a 45-pound beaver will defend 170 square yards,
write an equation for the area a defended by a
beaver weighing w pounds.
b. Scientists believe that thousands of years ago, the
beaver’s ancestors were 11 feet long and weighed
430 pounds. Use your equation to determine the area
defended by these animals.
ANSWER:
SOLUTION:
a.
Substitute 170 for y, 45 for b, and 1.31 for x in the
exponential function.
30. SCIENCEA mug of hot chocolate is 90°C at time t
= 0. It is surrounded by air at a constant temperature
of 20°C. If stirred steadily, its temperature in Celsius
−t
after t minutes will be y(t) = 20 + 70(1.071) .
a. Find the temperature of the hot chocolate after 15
minutes.
b. Find the temperature of the hot chocolate after 30 The equation for the area a defended by a beaver
minutes. weighting w pounds is
c.
The optimum drinking temperature is 60°C. Will b.
the mug of hot chocolate be at or below this Substitute 430 for w in the equation and solve for y.
temperature after 10 minutes?
SOLUTION:
a.
Substitute 15 for t in the equation and simplify.
ANSWER:
a 1.31
b. . a = 1.16w
b 2
Substitute 30 for t in the equation and simplify. . about 3268 yd
Solve each equation.
32.
c.
Substitute 10 for t in the equation and simplify.
SOLUTION:
So, temperature of the hot chocolate will be below
60°C after 10 minutes.
ANSWER:
a
. 45.02° C
b
. 28.94° C Use the Property of Equality for Exponential
c. below Functions.
31. ANIMALSStudies show that an animal will defend
a territory, with area in square yards, that is directly
proportional to the 1.31 power of the animal’s weight
in pounds.
a. If a 45-pound beaver will defend 170 square yards,
write an equation for the area a defended by a
beaver weighing w pounds. ANSWER:
b. Scientists believe that thousands of years ago, the
beaver’s ancestors were 11 feet long and weighed
430 pounds. Use your equation to determine the area
defended by these animals.
33.
SOLUTION:
a.
Substitute 170 for y, 45 for b, and 1.31 for x in the
exponential function. SOLUTION:
The equation for the area a defended by a beaver Use the Property of Equality for Exponential
weighting w pounds is Functions.
b.
Substitute 430 for w in the equation and solve for y.
ANSWER:
ANSWER:
a 1.31
. a = 1.16w
b 2
. about 3268 yd
Solve each equation. 34.
32.
SOLUTION:
SOLUTION:
Use the Property of Equality for Exponential
Functions.
Use the Property of Equality for Exponential
Functions.
ANSWER:
−6
35.
ANSWER:
SOLUTION:
33.
SOLUTION:
Use the Property of Equality for Exponential
Functions.
Use the Property of Equality for Exponential
Functions.
ANSWER:
36.
ANSWER:
SOLUTION:
34.
SOLUTION:
Use the Property of Equality for Exponential
Functions.
Use the Property of Equality for Exponential
Functions.
ANSWER:
−6 ANSWER:
35.
37.
SOLUTION:
SOLUTION:
Use the Property of Equality for Exponential Use the Property of Equality for Exponential
Functions. Functions.
ANSWER:
1
ANSWER:
38. CCSS MODELINGIn 1950, the world population
was about 2.556 billion. By 1980, it had increased to
about 4.458 billion.
a. x
Write an exponential function of the form y = ab
36. that could be used to model the world population y in
billions for 1950 to 1980. Write the equation in terms
SOLUTION: of x, the number of years since 1950. (Round the
value of b to the nearest ten-thousandth.)
b. Suppose the population continued to grow at that
rate. Estimate the population in 2000.
c. In 2000, the population of the world was about
6.08 billion. Compare your estimate to the actual
population.
d. Use the equation you wrote in Part a to estimate
the world population in the year 2020. How accurate
do you think the estimate is? Explain your reasoning.
Use the Property of Equality for Exponential
Functions. SOLUTION:
a.
Substitute 4.458 for y, 2.556 for a, and 30 for x in the
exponential function and solve for b.
ANSWER:
The exponential function that model the situation is
.
b.
37. Substitute 50 for x in the equation and simplify.
SOLUTION:
c. The prediction was about 375 million greater than
the actual population.
d.
Use the Property of Equality for Exponential Substitute 70 for x in the equation and simplify.
Functions.
Because the prediction for 2000 was greater than the
actual population, this prediction for 2020 is probably
ANSWER: even higher than the actual population will be at the
1
time.
38. CCSS MODELINGIn 1950, the world population
ANSWER:
was about 2.556 billion. By 1980, it had increased to x
a
about 4.458 billion. . y = 2.556(1.0187)
b
a. x . 6.455 billion
Write an exponential function of the form y = ab c. The prediction was about 375 million greater than
that could be used to model the world population y in the actual.
billions for 1950 to 1980. Write the equation in terms d
of x, the number of years since 1950. (Round the . About 9.3498 billion; because the prediction for
value of b to the nearest ten-thousandth.) 2000 was greater than the actual population, this
b. Suppose the population continued to grow at that prediction is probably even higher than the actual
rate. Estimate the population in 2000. population will be at the time.
c. In 2000, the population of the world was about 39. TREES The diameter of the base of a tree trunk in
6.08 billion. Compare your estimate to the actual centimeters varies directly with the power of its
population.
d. Use the equation you wrote in Part a to estimate height in meters.
the world population in the year 2020. How accurate a. A young sequoia tree is 6 meters tall, and the
do you think the estimate is? Explain your reasoning. diameter of its base is 19.1 centimeters. Use this
information to write an equation for the diameter d of
SOLUTION: the base of a sequoia tree if its height is h meters
a.
Substitute 4.458 for y, 2.556 for a, and 30 for x in the high
exponential function and solve for b. b. The General Sherman Tree in Sequoia National
Park, California, is approximately 84 meters tall.
Find the diameter of the General Sherman Tree at its
base.
SOLUTION:
The exponential function that model the situation is a.
. The equation that represent the situation is
.
b. b.
Substitute 50 for x in the equation and simplify. Substitute 84 for h in the equation and solve for d.
c. The prediction was about 375 million greater than
the actual population. The diameter of the General Sherman Tree at its
d. base is about 1001 cm.
Substitute 70 for x in the equation and simplify.
ANSWER:
a.
b
. about 1001 cm
Because the prediction for 2000 was greater than the 40. FINANCIAL LITERACYMrs. Jackson has two
actual population, this prediction for 2020 is probably different retirement investment plans from which to
even higher than the actual population will be at the choose.
time. a. Write equations for Option A and Option B given
the minimum deposits.
ANSWER: b. Draw a graph to show the balances for each
x
a investment option after t years.
. y = 2.556(1.0187)
b c. Explain whether Option A or Option B is the
. 6.455 billion better investment choice.
c. The prediction was about 375 million greater than
the actual.
d
. About 9.3498 billion; because the prediction for
2000 was greater than the actual population, this
prediction is probably even higher than the actual
population will be at the time.
39. TREES The diameter of the base of a tree trunk in
centimeters varies directly with the power of its
height in meters.
a. A young sequoia tree is 6 meters tall, and the SOLUTION:
diameter of its base is 19.1 centimeters. Use this a.
information to write an equation for the diameter d of Use the compound interest formula.
the base of a sequoia tree if its height is h meters The equation that represents Option A
high is .
b. The General Sherman Tree in Sequoia National
Park, California, is approximately 84 meters tall. The equation that represents Option B
Find the diameter of the General Sherman Tree at its
is
base.
b.
SOLUTION: The graph that shows the balances for each
a. investment option after t years:
The equation that represent the situation is
.
b.
Substitute 84 for h in the equation and solve for d.
The diameter of the General Sherman Tree at its
base is about 1001 cm.
ANSWER:
a. c.
b During the first 22 years, Option B is the better
. about 1001 cm choice because the total is greater than that of
Option A. However, after about 22 years, the
40. FINANCIAL LITERACYMrs. Jackson has two
different retirement investment plans from which to balance of Option A exceeds that of Option B, so
choose. Option A is the better choice.
a. Write equations for Option A and Option B given
the minimum deposits. ANSWER:
b. Draw a graph to show the balances for each a.
investment option after t years. b.
c. Explain whether Option A or Option B is the
better investment choice.
SOLUTION:
a.
Use the compound interest formula.
Sample answer:
The equation that represents Option A c. During the first 22 years, Option
B is the better choice because the total is greater
is . than that of Option A. However, after about 22
The equation that represents Option B years, the balance of Option A exceeds that of
Option B, so Option A is the better choice.
is 41. MULTIPLE REPRESENTATIONSIn this
problem, you will explore the rapid increase of an
b. exponential function. A large sheet of paper is cut in
The graph that shows the balances for each half, and one of the resulting pieces is placed on top
investment option after t years: of the other. Then the pieces in the stack are cut in
half and placed on top of each other. Suppose this
procedure is repeated several times.
a. CONCRETE
Perform this activity and count the
number of sheets in the stack after the first cut. How
many pieces will there be after the second cut? How
many pieces after the third cut? How many pieces
after the fourth cut?
b. TABULAR
Record your results in a table.
c. SYMBOLIC
Use the pattern in the table to write
an equation for the number of pieces in the stack
after x cuts.
d. ANALYTICAL
The thickness of ordinary paper
is about 0.003 inch. Write an equation for the
thickness of the stack of paper after x cuts.
e.ANALYTICAL
How thick will the stack of
c.
During the first 22 years, Option B is the better paper be after 30 cuts?
choice because the total is greater than that of
Option A. However, after about 22 years, the SOLUTION:
balance of Option A exceeds that of Option B, so a.
Option A is the better choice. There will be 2, 4, 8, 16 pieces after the first, second,
third and fourth cut respectively.
b.
ANSWER:
a.
b.
c.
The equation that represent the situation is
d.
Substitute 0.003 for a and 2 for b in the exponential
function.
e.
Substitute 30 for x in the equation and
simplify.
Sample answer:
c. During the first 22 years, Option
B is the better choice because the total is greater
than that of Option A. However, after about 22
years, the balance of Option A exceeds that of
Option B, so Option A is the better choice. The thickness of the stack of paper after 30 cuts is
about 3221225.47 in.
41. MULTIPLE REPRESENTATIONSIn this
problem, you will explore the rapid increase of an
exponential function. A large sheet of paper is cut in ANSWER:
half, and one of the resulting pieces is placed on top a. 2, 4, 8, 16
b
of the other. Then the pieces in the stack are cut in .
half and placed on top of each other. Suppose this
procedure is repeated several times.
a. CONCRETE
Perform this activity and count the
number of sheets in the stack after the first cut. How
many pieces will there be after the second cut? How
many pieces after the third cut? How many pieces x
after the fourth cut? c. y = 2
x
d
b. TABULAR . y = 0.003(2)
Record your results in a table.
e. about 3,221,225.47 in.
c. SYMBOLIC
Use the pattern in the table to write
an equation for the number of pieces in the stack
42. WRITING IN MATHIn a problem about
after x cuts. compound interest, describe what happens as the
d. ANALYTICAL
The thickness of ordinary paper compounding period becomes more frequent while
is about 0.003 inch. Write an equation for the the principal and overall time remain the same.
thickness of the stack of paper after x cuts.
e.ANALYTICAL
How thick will the stack of
SOLUTION:
paper be after 30 cuts? Sample answer: The more frequently interest is
compounded, the higher the account balance
SOLUTION: becomes.
a.
There will be 2, 4, 8, 16 pieces after the first, second,
third and fourth cut respectively. ANSWER:
b. Sample answer: The more frequently interest is
compounded, the higher the account balance
becomes.
x −
ERROR ANALYSIS
43. Beth and Liz are solving 6
3 > 36−x − 1. Is either of them correct? Explain your
reasoning.
c.
The equation that represent the situation is
d.
Substitute 0.003 for a and 2 for b in the exponential
function.
e.
Substitute 30 for x in the equation and
simplify.
The thickness of the stack of paper after 30 cuts is
about 3221225.47 in.
ANSWER:
a. 2, 4, 8, 16
b
.
SOLUTION:
Sample answer: Beth; Liz added the exponents
x instead of multiplying them when taking the power of
c. y = 2
x a power.
d
. y = 0.003(2)
e. about 3,221,225.47 in. ANSWER:
Sample answer: Beth; Liz added the exponents
42. WRITING IN MATHIn a problem about instead of multiplying them when taking the power of
compound interest, describe what happens as the
a power.
compounding period becomes more frequent while
the principal and overall time remain the same. 18 18 18
44. CHALLENGESolve for x: 16 + 16 + 16 +
18 18 x
16 + 16 = 4 .
SOLUTION:
Sample answer: The more frequently interest is
compounded, the higher the account balance SOLUTION:
becomes.
ANSWER:
Sample answer: The more frequently interest is
compounded, the higher the account balance
becomes.
x −
ERROR ANALYSIS
43. Beth and Liz are solving 6
3 > 36−x − 1. Is either of them correct? Explain your
reasoning.
ANSWER:
37.1610
45. OPEN ENDEDWhat would be a more beneficial
change to a 5-year loan at 8% interest compounded
monthly: reducing the term to 4 years or reducing the
interest rate to 6.5%?
SOLUTION:
Reducing the term will be more beneficial. The
multiplier is 1.3756 for the 4-year and 1.3828 for the
6.5%.
ANSWER:
Reducing the term will be more beneficial. The
multiplier is 1.3756 for the 4-year and 1.3828 for the
6.5%.
CCSS ARGUMENTS
46. Determine whether the
following statements are sometimes, always, or
never true. Explain your reasoning.
a. x 20x
2 > −8 for all values of x.
b. The graph of an exponential growth equation is
increasing.
c. The graph of an exponential decay equation is
increasing.
SOLUTION:
Sample answer: Beth; Liz added the exponents SOLUTION:
a. x 20x
instead of multiplying them when taking the power of Always; 2 will always be positive, and −8 will
always be negative.
a power.
b. Always; by definition the graph will always be
ANSWER: increasing even if it is a small increase.
Sample answer: Beth; Liz added the exponents
c. Never; by definition the graph will always be
instead of multiplying them when taking the power of decreasing even if it is a small decrease.
a power.
ANSWER:
18 18 18 x 20x
44. CHALLENGESolve for x: 16 + 16 + 16 + a
. Always; 2 will always be positive, and 8 will
18 18 x −
16 + 16 = 4 . always be negative.
b
. Always; by definition the graph will always be
SOLUTION: increasing even if it is a small increase.
c. Never; by definition the graph will always be
decreasing even if it is a small decrease.
OPEN ENDEDWrite an exponential inequality with
47.
a solution of x 2.
≤
SOLUTION:
x 2
Sample answer: 4 4
≤
ANSWER:
x 2
ANSWER: Sample answer: 4 ≤ 4
37.1610 2x x + 1 2x + 2 4x + 1
PROOFShow that 27 · 81 = 3 · 9 .
48.
45. OPEN ENDEDWhat would be a more beneficial
change to a 5-year loan at 8% interest compounded SOLUTION:
monthly: reducing the term to 4 years or reducing the
interest rate to 6.5%?
SOLUTION:
Reducing the term will be more beneficial. The
multiplier is 1.3756 for the 4-year and 1.3828 for the
6.5%.
ANSWER:
Reducing the term will be more beneficial. The
multiplier is 1.3756 for the 4-year and 1.3828 for the
6.5%.
ANSWER:
CCSS ARGUMENTS
46. Determine whether the
following statements are sometimes, always, or
never true. Explain your reasoning.
a. x 20x
2 > −8 for all values of x.
b. The graph of an exponential growth equation is
increasing.
c. The graph of an exponential decay equation is
increasing.
WRITING IN MATHIf you were given the initial
49.
SOLUTION: and final amounts of a radioactive substance and the
a. x 20x amount of time that passes, how would you
Always; 2 will always be positive, and −8 will
always be negative. determine the rate at which the amount was
b. Always; by definition the graph will always be increasing or decreasing in order to write an
increasing even if it is a small increase. equation?
c. Never; by definition the graph will always be
decreasing even if it is a small decrease. SOLUTION:
Sample answer: Divide the final amount by the initial
amount. If n is the number of time intervals that pass,
ANSWER: take the nth root of the answer.
a x 20x
. Always; 2 will always be positive, and 8 will
−
always be negative.
ANSWER:
b
. Always; by definition the graph will always be Sample answer: Divide the final amount by the initial
increasing even if it is a small increase. amount. If n is the number of time intervals that pass,
c. Never; by definition the graph will always be take the nth root of the answer.
decreasing even if it is a small decrease.
−4 =
50. 3 × 10
OPEN ENDEDWrite an exponential inequality with
47. A 30,000
a solution of x 2. −
≤ B 0.0003
C 120
−
SOLUTION:
x 2 D0.00003
Sample answer: 4 4
≤
SOLUTION:
ANSWER:
x 2
Sample answer: 4 4
≤
2x x + 1 2x + 2 4x + 1
PROOFShow that 27 · 81 = 3 · 9 .
48.
SOLUTION:
B is the correct option.
ANSWER:
B
51. Which of the following could not be a solution to 5 −
3x < 3?
−
F2.5
G3
H 3.5
J
4
ANSWER:
SOLUTION:
Check the inequality by substituting 2.5 for x.
So, F is the correct option.
WRITING IN MATHIf you were given the initial
49.
and final amounts of a radioactive substance and the ANSWER:
F
amount of time that passes, how would you
determine the rate at which the amount was
GRIDDED RESPONSEThe three angles of a
increasing or decreasing in order to write an 52.
equation? triangle are 3x, x + 10, and 2x − 40. Find the measure
of the smallest angle in the triangle.
SOLUTION:
Sample answer: Divide the final amount by the initial SOLUTION:
Sum of the three angles in a triangle is 180 .
amount. If n is the number of time intervals that pass, º
take the nth root of the answer.
ANSWER:
Sample answer: Divide the final amount by the initial
amount. If n is the number of time intervals that pass,
take the nth root of the answer.
−4 =
50. 3 × 10
A 30,000
−
B 0.0003
C 120
−
D 0.00003
The measure of the smallest angle in the triangle is
30 .
º
SOLUTION:
ANSWER:
30
SAT/ACT Which of the following is equivalent to
53.
(x)(x)(x)(x) for all x?
A x + 4
B is the correct option. B 4
x
2
ANSWER: C 2x
B
D 4x2
Which of the following could not be a solution to 5 4
51. − E
3x < 3? x
−
SOLUTION:
F2.5
G3
H 3.5
J
4
E is the correct choice.
SOLUTION:
Check the inequality by substituting 2.5 for x.
ANSWER:
E
Graph each function.
x
y = 2(3)
54.
So, F is the correct option.
SOLUTION:
Make a table of values. Then plot the points and
ANSWER: sketch the graph.
F
GRIDDED RESPONSEThe three angles of a
52.
triangle are 3x, x + 10, and 2x − 40. Find the measure
of the smallest angle in the triangle.
SOLUTION:
Sum of the three angles in a triangle is 180 .
º
The measure of the smallest angle in the triangle is
30 .
º
ANSWER:
ANSWER:
30
SAT/ACT Which of the following is equivalent to
53.
(x)(x)(x)(x) for all x?
A x + 4
B 4x
C 2x2
D 4x2
E 4
x x
y = 5(2)
55.
SOLUTION:
SOLUTION:
Make a table of values. Then plot the points and
sketch the graph.
E is the correct choice.
ANSWER:
E
Graph each function.
x
y = 2(3)
54.
SOLUTION:
Make a table of values. Then plot the points and
sketch the graph.
ANSWER:
ANSWER:
56.
SOLUTION:
Make a table of values. Then plot the points and
sketch the graph.
x
y = 5(2)
55.
SOLUTION:
Make a table of values. Then plot the points and
sketch the graph.
ANSWER:
ANSWER: Solve each equation.
57.
SOLUTION:
ANSWER:
4
56.
SOLUTION:
Make a table of values. Then plot the points and 58.
sketch the graph. SOLUTION:
ANSWER:
18
59.
SOLUTION:
ANSWER:
ANSWER:
8.5
60.
SOLUTION:
Solve each equation.
57.
The square root of x cannot be negative, so there is
SOLUTION: no solution.
ANSWER:
no solution
61.
SOLUTION:
ANSWER:
4
58.
SOLUTION:
ANSWER:
5
62.
SOLUTION:
ANSWER:
18
59.
SOLUTION:
ANSWER:
20
−
63.
ANSWER:
8.5
SOLUTION:
60.
SOLUTION:
ANSWER:
5
The square root of x cannot be negative, so there is
no solution. 64.
SOLUTION:
ANSWER:
no solution
61.
SOLUTION:
ANSWER:
ANSWER:
5
65.
62.
SOLUTION:
SOLUTION:
ANSWER:
1
−
ANSWER:
20
− SALES A salesperson earns $10 an hour plus a 10%
66.
commission on sales. Write a function to describe the
salesperson’s income. If the salesperson wants to
63. earn $1000 in a 40-hour week, what should his sales
be?
SOLUTION:
SOLUTION:
Let I be the income of the salesperson and m be his
sales.
The function that represent the situation is
.
Substitute 1000 for I in the equation and solve for m.
ANSWER:
5
ANSWER:
I(m) = 400 + 0.1m; $6000
64.
STATE FAIRA dairy makes three types of
SOLUTION: 67.
cheese cheddar, Monterey Jack, and Swiss and
— —
sells the cheese in three booths at the state fair. At
the beginning of one day, the first booth received x
pounds of each type of cheese. The second booth
received y pounds of each type of cheese, and the
third booth received z pounds of each type of cheese.
By the end of the day, the dairy had sold 131 pounds
of cheddar, 291 pounds of Monterey Jack, and 232
pounds of Swiss. The table below shows the percent
of the cheese delivered in the morning that was sold
at each booth. How many pounds of cheddar cheese
ANSWER: did each booth receive in the morning?
65.
SOLUTION:
SOLUTION:
The system of equations that represent the situation:
ANSWER: Eliminate the variable x by using two pairs of
1 equations.
−
SALESA salesperson earns $10 an hour plus a 10%
66. Subtract (1) and (2).
commission on sales. Write a function to describe the
salesperson s income. If the salesperson wants to
’
earn $1000 in a 40-hour week, what should his sales
be?
SOLUTION:
Let I be the income of the salesperson and m be his
Multiply (2) by 3 and (3) by 4 and subtract both the
sales. equations.
The function that represent the situation is
.
Substitute 1000 for I in the equation and solve for m.
Solve the system of two equations:
ANSWER:
I(m) = 400 + 0.1m; $6000
STATE FAIRA dairy makes three types of
67.
cheese cheddar, Monterey Jack, and Swiss and
— —
sells the cheese in three booths at the state fair. At
the beginning of one day, the first booth received x
pounds of each type of cheese. The second booth
received y pounds of each type of cheese, and the Substitute z = 100 in the equation
third booth received z pounds of each type of cheese.
By the end of the day, the dairy had sold 131 pounds
of cheddar, 291 pounds of Monterey Jack, and 232
pounds of Swiss. The table below shows the percent
of the cheese delivered in the morning that was sold
at each booth. How many pounds of cheddar cheese
did each booth receive in the morning?
Substitute y = 150 and z = 100 in the (1) and solve
for x.
SOLUTION:
The system of equations that represent the situation:
Booth 1 has 190 lb; Booth 2 has 150 lb; Booth 3 has
100 lb.
ANSWER:
booth 1, 190 lb; booth 2, 150 lb; booth 3, 100 lb
Find [g h](x) and [h g](x).
Eliminate the variable x by using two pairs of ◦ ◦
68. h(x) = 2x − 1
equations.
g(x) = 3x + 4
Subtract (1) and (2).
SOLUTION:
Multiply (2) by 3 and (3) by 4 and subtract both the
equations.
ANSWER:
6x + 1; 6x + 7
h(x) = x2 + 2
69.
Solve the system of two equations: g(x) = x − 3
SOLUTION:
Substitute z = 100 in the equation
ANSWER:
x2 1; x2 6x + 11
− −
h(x) = x2 + 1
70.
g(x) = 2x + 1
−
Substitute y = 150 and z = 100 in the (1) and solve SOLUTION:
for x.
Booth 1 has 190 lb; Booth 2 has 150 lb; Booth 3 has ANSWER:
2x2 1; 4x2 4x + 2
100 lb. − − −
h(x) = 5x
71. −
ANSWER:
booth 1, 190 lb; booth 2, 150 lb; booth 3, 100 lb g(x) = 3x − 5
Find [g h](x) and [h g](x). SOLUTION:
◦ ◦
68. h(x) = 2x − 1
g(x) = 3x + 4
SOLUTION:
ANSWER:
15x 5; 15x + 25
− − −
h(x) = x3
72.
g(x) = x − 2
SOLUTION:
ANSWER:
6x + 1; 6x + 7
h(x) = x2 + 2
69.
g(x) = x − 3
ANSWER:
3 3 2
SOLUTION:
x x 6x + 12x 8
−2; − −
73. h(x) = x + 4
g(x) = | x |
SOLUTION:
ANSWER:
2 2 ANSWER:
x 1; x 6x + 11
− − | x + 4 | ; | x | + 4
h(x) = x2 + 1
70.
g(x) = 2x + 1
−
SOLUTION:
ANSWER:
2x2 1; 4x2 4x + 2
− − −
h(x) = 5x
71. −
g(x) = 3x − 5
SOLUTION:
ANSWER:
15x 5; 15x + 25
− − −
h(x) = x3
72.
g(x) = x − 2
SOLUTION:
ANSWER:
3 3 2
x x 6x + 12x 8
−2; − −
73. h(x) = x + 4
g(x) = | x |
SOLUTION:
ANSWER:
| x + 4 | ; | x | + 4
Solve each equation.
5x 2x − 4
1. 3 = 27
SOLUTION:
Use the Property of Equality for Exponential
Functions.
ANSWER:
12
2y − 3 y + 1
Solve each equation. 2. 16 = 4
5x 2x − 4
1. 3 = 27
SOLUTION:
SOLUTION:
Use the Property of Equality for Exponential
Use the Property of Equality for Exponential Functions.
Functions.
ANSWER:
12
ANSWER:
2y − 3 y + 1
2. 16 = 4
SOLUTION: 6x x − 2
3. 2 = 32
SOLUTION:
Use the Property of Equality for Exponential
Functions.
Use the Property of Equality for Exponential
Functions.
ANSWER:
ANSWER:
−10
x + 5 8x − 6
4. 49 = 7
6x x − 2
3. 2 = 32
SOLUTION:
SOLUTION:
Use the Property of Equality for Exponential
Use the Property of Equality for Exponential Functions.
Functions.
ANSWER:
−10 ANSWER:
x + 5 8x − 6
4. 49 = 7
SOLUTION: SCIENCE
5. Mitosis is a process in which one cell
divides into two. The Escherichia coli is one of the
fastest growing bacteria. It can reproduce itself in 15
minutes.
a. Write an exponential function to represent the
number of cells c after t minutes.
Use the Property of Equality for Exponential b. If you begin with one Escherichia coli cell, how
Functions. many cells will there be in one hour?
SOLUTION:
a.
The exponential function that represent the number
of cells after t minutes is .
b.
Substitute 1 for t in the function and solve for c.
ANSWER:
ANSWER:
SCIENCE
5. Mitosis is a process in which one cell
divides into two. The Escherichia coli is one of the a.
fastest growing bacteria. It can reproduce itself in 15 b. 16 cells
minutes.
a. Write an exponential function to represent the 6. A certificate of deposit (CD) pays 2.25% annual
number of cells c after t minutes. interest compounded biweekly. If you deposit $500
b. If you begin with one Escherichia coli cell, how into this CD, what will the balance be after 6 years?
many cells will there be in one hour?
SOLUTION:
SOLUTION: Use the compound interest formula.
a. Substitute $500 for P, 0.0225 for r, 26 for n and 6 for
The exponential function that represent the number t and simplify.
of cells after t minutes is .
b.
Substitute 1 for t in the function and solve for c.
ANSWER:
a. ANSWER:
b. 16 cells $572.23
6. A certificate of deposit (CD) pays 2.25% annual Solve each inequality.
interest compounded biweekly. If you deposit $500 2x + 6 2x – 4
into this CD, what will the balance be after 6 years? 7. 4 ≤64
SOLUTION:
SOLUTION:
Use the compound interest formula.
Substitute $500 for P, 0.0225 for r, 26 for n and 6 for
t and simplify.
Use the Property of Inequality for Exponential
Functions.
ANSWER:
$572.23 ANSWER:
x ≥ 4.5
Solve each inequality.
2x + 6 2x – 4
7. 4 ≤64
8.
SOLUTION:
SOLUTION:
Use the Property of Inequality for Exponential
Functions.
Use the Property of Inequality for Exponential
Functions.
ANSWER:
ANSWER:
x ≥ 4.5
Solve each equation.
8. 4x + 2
9. 8 = 64
SOLUTION:
SOLUTION:
Use the Property of Equality for Exponential
Functions.
Use the Property of Inequality for Exponential
Functions.
ANSWER:
0
ANSWER:
x − 6
10. 5 = 125
Solve each equation.
4x + 2 SOLUTION:
9. 8 = 64
SOLUTION:
Use the Property of Equality for Exponential
Functions.
Use the Property of Equality for Exponential
Functions.
ANSWER:
9
ANSWER: a + 2 3a + 1
0 11. 81 = 3
SOLUTION:
x − 6
10. 5 = 125
SOLUTION:
Use the Property of Equality for Exponential
Use the Property of Equality for Exponential Functions.
Functions.
ANSWER:
−7
ANSWER:
9
b + 2 2 − 2b
12. 256 = 4
a + 2 3a + 1
11. 81 = 3
SOLUTION:
SOLUTION:
Use the Property of Equality for Exponential
Use the Property of Equality for Exponential Functions.
Functions.
ANSWER:
ANSWER:
−1
−7
b + 2 2 − 2b 3c + 1 3c − 1
12. 256 = 4 13. 9 = 27
SOLUTION:
SOLUTION:
Use the Property of Equality for Exponential Use the Property of Equality for Exponential
Functions. Functions.
ANSWER:
−1 ANSWER:
3c + 1 3c − 1
13. 9 = 27
SOLUTION: 2y + 4 y + 1
14. 8 = 16
SOLUTION:
Use the Property of Equality for Exponential
Functions.
Use the Property of Equality for Exponential
Functions.
ANSWER:
7-2 Solving Exponential Equations and Inequalities ANSWER:
−4
2y + 4 y + 1
14. 8 = 16 15. CCSS MODELINGIn 2009, My-Lien received
$10,000 from her grandmother. Her parents invested
SOLUTION: all of the money, and by 2021, the amount will have
grown to $16,960.
a. Write an exponential function that could be used to
model the money y. Write the function in terms of x,
the number of years since 2009.
b. Assume that the amount of money continues to
Use the Property of Equality for Exponential grow at the same rate. What would be the balance in
Functions. the account in 2031?
SOLUTION:
a.
Substitute 16780 for y 10000 for a and 12 for x in the
exponential function and simplify.
ANSWER:
−4
15. CCSS MODELINGIn 2009, My-Lien received
$10,000 from her grandmother. Her parents invested
all of the money, and by 2021, the amount will have
grown to $16,960. The exponential function that models the situation
a. Write an exponential function that could be used to
model the money y. Write the function in terms of x, is .
the number of years since 2009.
b. Assume that the amount of money continues to b.
grow at the same rate. What would be the balance in Substitute 22 for x in the modeled function and solve
the account in 2031? for y.
SOLUTION:
a.
Substitute 16780 for y 10000 for a and 12 for x in the
exponential function and simplify.
ANSWER:
x
a
. y = 10,000(1.045)
b
. about $26,336.52
Write an exponential function for the graph that
passes through the given points.
16. (0, 6.4) and (3, 100)
The exponential function that models the situation
SOLUTION:
is . Substitute 100 for y and 6.4 for a and 3 for x into an
exponential function and determine the value of b.
b.
Substitute 22 for x in the modeled function and solve
for y.
eSolutions Manual - Powered by Cognero Page4
ANSWER: An exponential function that passes through the given
x
a
. y = 10,000(1.045) points is .
b
. about $26,336.52
Write an exponential function for the graph that ANSWER:
x
passes through the given points. y = 6.4(2.5)
16. (0, 6.4) and (3, 100)
SOLUTION: 17. (0, 256) and (4, 81)
Substitute 100 for y and 6.4 for a and 3 for x into an
exponential function and determine the value of b. SOLUTION:
Substitute 81 for y and 256 for a and 4 for x into an
exponential function and determine the value of b.
An exponential function that passes through the given
points is . An exponential function that passes through the given
points is .
ANSWER:
x
y = 6.4(2.5)
ANSWER:
x
y = 256(0.75)
17. (0, 256) and (4, 81)
18. (0, 128) and (5, 371,293)
SOLUTION:
Substitute 81 for y and 256 for a and 4 for x into an SOLUTION:
exponential function and determine the value of b. Substitute 371293 for y and 128 for a and 5 for x into
an exponential function and determine the value of b.
An exponential function that passes through the given
points is . An exponential function that passes through the given
points is .
ANSWER:
x
y = 256(0.75) ANSWER:
x
y = 128(4.926)
18. (0, 128) and (5, 371,293)
19. (0, 144), and (4, 21,609)
SOLUTION:
Substitute 371293 for y and 128 for a and 5 for x into SOLUTION:
an exponential function and determine the value of b. Substitute 21609 for y and 144 for a and 4 for x into
an exponential function and determine the value of b.
An exponential function that passes through the given
points is . An exponential function that passes through the given
points is .
ANSWER:
x
y = 128(4.926) ANSWER:
x
y = 144(3.5)
19. (0, 144), and (4, 21,609)
20. Find the balance of an account after 7 years if $700
SOLUTION: is deposited into an account paying 4.3% interest
Substitute 21609 for y and 144 for a and 4 for x into compounded monthly.
an exponential function and determine the value of b.
SOLUTION:
Use the compound interest formula.
Substitute $700 for P, 0.043 for r, 12 for n and 7 for t
and simplify.
An exponential function that passes through the given
points is .
ANSWER:
ANSWER:
x
y = 144(3.5) $945.34
20. Find the balance of an account after 7 years if $700 21. Determine how much is in a retirement account after
is deposited into an account paying 4.3% interest 20 years if $5000 was invested at 6.05% interest
compounded monthly. compounded weekly.
SOLUTION:
SOLUTION:
Use the compound interest formula. Use the compound interest formula.
Substitute $700 for P, 0.043 for r, 12 for n and 7 for t Substitute $5000 for P, 0.0605 for r, 52 for n and 20
and simplify. for t and simplify.
ANSWER:
ANSWER:
$945.34 $16,755.63
21. Determine how much is in a retirement account after 22. A savings account offers 0.7% interest compounded
20 years if $5000 was invested at 6.05% interest bimonthly. If $110 is deposited in this account, what
compounded weekly. will the balance be after 15 years?
SOLUTION:
SOLUTION:
Use the compound interest formula. Use the compound interest formula.
Substitute $5000 for P, 0.0605 for r, 52 for n and 20 Substitute $110 for P, 0.007 for r, 6 for n and 15 for t
for t and simplify. and simplify.
ANSWER:
ANSWER:
$16,755.63 $122.17
22. A savings account offers 0.7% interest compounded 23. A college savings account pays 13.2% annual
bimonthly. If $110 is deposited in this account, what interest compounded semiannually. What is the
will the balance be after 15 years? balance of an account after 12 years if $21,000 was
initially deposited?
SOLUTION:
Use the compound interest formula. SOLUTION:
Substitute $110 for P, 0.007 for r, 6 for n and 15 for t Use the compound interest formula.
and simplify. Substitute $21,000 for P, 0.132 for r, 2 for n and 12
for t and simplify.
ANSWER:
$122.17 ANSWER:
23. A college savings account pays 13.2% annual $97,362.61
interest compounded semiannually. What is the Solve each inequality.
balance of an account after 12 years if $21,000 was
initially deposited? 24.
SOLUTION:
SOLUTION:
Use the compound interest formula.
Substitute $21,000 for P, 0.132 for r, 2 for n and 12
for t and simplify.
Use the Property of Inequality for Exponential
Functions.
ANSWER:
ANSWER:
$97,362.61
25.
Solve each inequality.
SOLUTION:
24.
SOLUTION:
Use the Property of Inequality for Exponential
Functions.
Use the Property of Inequality for Exponential
Functions.
ANSWER:
ANSWER:
25.
26.
SOLUTION:
SOLUTION:
Use the Property of Inequality for Exponential
Functions.
Use the Property of Inequality for Exponential
Functions.
ANSWER:
26.
ANSWER:
SOLUTION:
27.
SOLUTION:
Use the Property of Inequality for Exponential
Functions.
Use the Property of Inequality for Exponential
Functions.
ANSWER:
ANSWER:
27.
SOLUTION: 28.
SOLUTION:
Use the Property of Inequality for Exponential
Functions.
Use the Property of Inequality for Exponential
Functions.
ANSWER:
28. ANSWER:
SOLUTION:
29.
SOLUTION:
Use the Property of Inequality for Exponential
Functions.
Use the Property of Inequality for Exponential
Functions.
ANSWER:
29.
ANSWER:
SOLUTION:
30. SCIENCEA mug of hot chocolate is 90°C at time t
= 0. It is surrounded by air at a constant temperature
of 20°C. If stirred steadily, its temperature in Celsius
−t
after t minutes will be y(t) = 20 + 70(1.071) .
a. Find the temperature of the hot chocolate after 15
minutes.
Use the Property of Inequality for Exponential b. Find the temperature of the hot chocolate after 30
Functions. minutes.
c.
The optimum drinking temperature is 60°C. Will
the mug of hot chocolate be at or below this
temperature after 10 minutes?
SOLUTION:
a.
Substitute 15 for t in the equation and simplify.
ANSWER:
b.
30. SCIENCEA mug of hot chocolate is 90°C at time t Substitute 30 for t in the equation and simplify.
= 0. It is surrounded by air at a constant temperature
of 20°C. If stirred steadily, its temperature in Celsius
−t
after t minutes will be y(t) = 20 + 70(1.071) .
a. Find the temperature of the hot chocolate after 15 c.
minutes. Substitute 10 for t in the equation and simplify.
b. Find the temperature of the hot chocolate after 30
minutes.
c. So, temperature of the hot chocolate will be below
The optimum drinking temperature is 60°C. Will
the mug of hot chocolate be at or below this 60°C after 10 minutes.
temperature after 10 minutes?
ANSWER:
SOLUTION: a
a. . 45.02° C
b
Substitute 15 for t in the equation and simplify. . 28.94° C
c. below
31. ANIMALSStudies show that an animal will defend
b. a territory, with area in square yards, that is directly
Substitute 30 for t in the equation and simplify. proportional to the 1.31 power of the animal’s weight
in pounds.
a. If a 45-pound beaver will defend 170 square yards,
write an equation for the area a defended by a
beaver weighing w pounds.
c. b. Scientists believe that thousands of years ago, the
Substitute 10 for t in the equation and simplify. beaver’s ancestors were 11 feet long and weighed
430 pounds. Use your equation to determine the area
defended by these animals.
So, temperature of the hot chocolate will be below
SOLUTION:
60°C after 10 minutes. a.
ANSWER: Substitute 170 for y, 45 for b, and 1.31 for x in the
a exponential function.
. 45.02° C
b
. 28.94° C
c. below
31. ANIMALSStudies show that an animal will defend
a territory, with area in square yards, that is directly
proportional to the 1.31 power of the animal’s weight
in pounds. The equation for the area a defended by a beaver
a. If a 45-pound beaver will defend 170 square yards, weighting w pounds is
write an equation for the area a defended by a b.
beaver weighing w pounds. Substitute 430 for w in the equation and solve for y.
b. Scientists believe that thousands of years ago, the
beaver’s ancestors were 11 feet long and weighed
430 pounds. Use your equation to determine the area
defended by these animals.
SOLUTION:
a. ANSWER:
a 1.31
Substitute 170 for y, 45 for b, and 1.31 for x in the . a = 1.16w
exponential function. b 2
. about 3268 yd
Solve each equation.
32.
SOLUTION:
The equation for the area a defended by a beaver
weighting w pounds is
b.
Substitute 430 for w in the equation and solve for y.
Use the Property of Equality for Exponential
Functions.
ANSWER:
a 1.31
. a = 1.16w
b 2
. about 3268 yd
Solve each equation.
ANSWER:
32.
SOLUTION:
33.
SOLUTION:
Use the Property of Equality for Exponential
Functions.
Use the Property of Equality for Exponential
Functions.
ANSWER:
ANSWER:
33.
SOLUTION:
34.
SOLUTION:
Use the Property of Equality for Exponential
Functions.
Use the Property of Equality for Exponential
Functions.
ANSWER:
ANSWER:
−6
34.
35.
SOLUTION:
SOLUTION:
Use the Property of Equality for Exponential
Functions.
Use the Property of Equality for Exponential
Functions.
ANSWER:
−6
35.
ANSWER:
SOLUTION:
36.
SOLUTION:
Use the Property of Equality for Exponential
Functions.
Use the Property of Equality for Exponential
ANSWER: Functions.
36.
SOLUTION:
ANSWER:
37.
SOLUTION:
Use the Property of Equality for Exponential
Functions.
Use the Property of Equality for Exponential
Functions.
ANSWER:
ANSWER:
1
37.
38. CCSS MODELINGIn 1950, the world population
SOLUTION: was about 2.556 billion. By 1980, it had increased to
about 4.458 billion.
a. x
Write an exponential function of the form y = ab
that could be used to model the world population y in
billions for 1950 to 1980. Write the equation in terms
of x, the number of years since 1950. (Round the
value of b to the nearest ten-thousandth.)
Use the Property of Equality for Exponential b. Suppose the population continued to grow at that
Functions. rate. Estimate the population in 2000.
c. In 2000, the population of the world was about
6.08 billion. Compare your estimate to the actual
population.
d. Use the equation you wrote in Part a to estimate
the world population in the year 2020. How accurate
do you think the estimate is? Explain your reasoning.
ANSWER:
1 SOLUTION:
a.
38. CCSS MODELINGIn 1950, the world population Substitute 4.458 for y, 2.556 for a, and 30 for x in the
was about 2.556 billion. By 1980, it had increased to exponential function and solve for b.
about 4.458 billion.
a. x
Write an exponential function of the form y = ab
that could be used to model the world population y in
billions for 1950 to 1980. Write the equation in terms
of x, the number of years since 1950. (Round the
value of b to the nearest ten-thousandth.)
b. Suppose the population continued to grow at that The exponential function that model the situation is
rate. Estimate the population in 2000. .
c. In 2000, the population of the world was about b.
6.08 billion. Compare your estimate to the actual Substitute 50 for x in the equation and simplify.
population.
d. Use the equation you wrote in Part a to estimate
the world population in the year 2020. How accurate
do you think the estimate is? Explain your reasoning.
SOLUTION:
a. c. The prediction was about 375 million greater than
Substitute 4.458 for y, 2.556 for a, and 30 for x in the the actual population.
exponential function and solve for b. d.
Substitute 70 for x in the equation and simplify.
The exponential function that model the situation is Because the prediction for 2000 was greater than the
. actual population, this prediction for 2020 is probably
even higher than the actual population will be at the
b. time.
Substitute 50 for x in the equation and simplify.
ANSWER:
x
a
. y = 2.556(1.0187)
b
. 6.455 billion
c. The prediction was about 375 million greater than
the actual.
c. The prediction was about 375 million greater than d
the actual population. . About 9.3498 billion; because the prediction for
d. 2000 was greater than the actual population, this
Substitute 70 for x in the equation and simplify. prediction is probably even higher than the actual
population will be at the time.
39. TREES The diameter of the base of a tree trunk in
centimeters varies directly with the power of its
height in meters.
Because the prediction for 2000 was greater than the a. A young sequoia tree is 6 meters tall, and the
actual population, this prediction for 2020 is probably diameter of its base is 19.1 centimeters. Use this
even higher than the actual population will be at the information to write an equation for the diameter d of
time. the base of a sequoia tree if its height is h meters
high
ANSWER: b. The General Sherman Tree in Sequoia National
x
a
. y = 2.556(1.0187) Park, California, is approximately 84 meters tall.
b
. 6.455 billion Find the diameter of the General Sherman Tree at its
c. The prediction was about 375 million greater than
the actual. base.
d
. About 9.3498 billion; because the prediction for
2000 was greater than the actual population, this SOLUTION:
prediction is probably even higher than the actual a.
population will be at the time. The equation that represent the situation is
39. TREES The diameter of the base of a tree trunk in .
centimeters varies directly with the power of its b.
Substitute 84 for h in the equation and solve for d.
height in meters.
a. A young sequoia tree is 6 meters tall, and the
diameter of its base is 19.1 centimeters. Use this
information to write an equation for the diameter d of
the base of a sequoia tree if its height is h meters
The diameter of the General Sherman Tree at its
high
b. The General Sherman Tree in Sequoia National base is about 1001 cm.
Park, California, is approximately 84 meters tall.
Find the diameter of the General Sherman Tree at its ANSWER:
base. a.
b
. about 1001 cm
SOLUTION:
a. 40. FINANCIAL LITERACYMrs. Jackson has two
The equation that represent the situation is different retirement investment plans from which to
choose.
. a. Write equations for Option A and Option B given
b. the minimum deposits.
Substitute 84 for h in the equation and solve for d. b. Draw a graph to show the balances for each
investment option after t years.
c. Explain whether Option A or Option B is the
better investment choice.
The diameter of the General Sherman Tree at its
base is about 1001 cm.
ANSWER:
a.
b
. about 1001 cm
40. FINANCIAL LITERACYMrs. Jackson has two
different retirement investment plans from which to SOLUTION:
choose. a.
a. Write equations for Option A and Option B given Use the compound interest formula.
the minimum deposits. The equation that represents Option A
b. Draw a graph to show the balances for each is .
investment option after t years. The equation that represents Option B
c. Explain whether Option A or Option B is the
better investment choice. is
b.
The graph that shows the balances for each
investment option after t years:
SOLUTION:
a.
Use the compound interest formula.
The equation that represents Option A
is .
The equation that represents Option B
is
c.
b. During the first 22 years, Option B is the better
The graph that shows the balances for each choice because the total is greater than that of
investment option after t years: Option A. However, after about 22 years, the
balance of Option A exceeds that of Option B, so
Option A is the better choice.
ANSWER:
a.
b.
c.
During the first 22 years, Option B is the better
choice because the total is greater than that of
Option A. However, after about 22 years, the
balance of Option A exceeds that of Option B, so
Option A is the better choice.
Sample answer:
c. During the first 22 years, Option
ANSWER: B is the better choice because the total is greater
a. than that of Option A. However, after about 22
b. years, the balance of Option A exceeds that of
Option B, so Option A is the better choice.
41. MULTIPLE REPRESENTATIONSIn this
problem, you will explore the rapid increase of an
exponential function. A large sheet of paper is cut in
half, and one of the resulting pieces is placed on top
of the other. Then the pieces in the stack are cut in
half and placed on top of each other. Suppose this
procedure is repeated several times.
a. CONCRETE
Perform this activity and count the
number of sheets in the stack after the first cut. How
many pieces will there be after the second cut? How
many pieces after the third cut? How many pieces
after the fourth cut?
b. TABULAR
Record your results in a table.
Sample answer:
c. c. SYMBOLIC
During the first 22 years, Option Use the pattern in the table to write
B is the better choice because the total is greater an equation for the number of pieces in the stack
than that of Option A. However, after about 22 after x cuts.
years, the balance of Option A exceeds that of d. ANALYTICAL
Option B, so Option A is the better choice. The thickness of ordinary paper
is about 0.003 inch. Write an equation for the
thickness of the stack of paper after x cuts.
41. MULTIPLE REPRESENTATIONSIn this
e.ANALYTICAL
problem, you will explore the rapid increase of an How thick will the stack of
exponential function. A large sheet of paper is cut in paper be after 30 cuts?
half, and one of the resulting pieces is placed on top SOLUTION:
of the other. Then the pieces in the stack are cut in a.
half and placed on top of each other. Suppose this There will be 2, 4, 8, 16 pieces after the first, second,
procedure is repeated several times. third and fourth cut respectively.
a. CONCRETE b.
Perform this activity and count the
number of sheets in the stack after the first cut. How
many pieces will there be after the second cut? How
many pieces after the third cut? How many pieces
after the fourth cut?
b. TABULAR
Record your results in a table.
c. SYMBOLIC
Use the pattern in the table to write
an equation for the number of pieces in the stack c.
after x cuts. The equation that represent the situation is
d. ANALYTICAL
The thickness of ordinary paper d.
is about 0.003 inch. Write an equation for the Substitute 0.003 for a and 2 for b in the exponential
thickness of the stack of paper after x cuts. function.
e.ANALYTICAL
How thick will the stack of
paper be after 30 cuts?
e.
SOLUTION:
a. Substitute 30 for x in the equation and
There will be 2, 4, 8, 16 pieces after the first, second, simplify.
third and fourth cut respectively.
b.
The thickness of the stack of paper after 30 cuts is
about 3221225.47 in.
ANSWER:
c. a. 2, 4, 8, 16
The equation that represent the situation is b
d. .
Substitute 0.003 for a and 2 for b in the exponential
function.
e.
Substitute 30 for x in the equation and x
c. y = 2
simplify. x
d
. y = 0.003(2)
e. about 3,221,225.47 in.
42. WRITING IN MATHIn a problem about
compound interest, describe what happens as the
The thickness of the stack of paper after 30 cuts is compounding period becomes more frequent while
about 3221225.47 in. the principal and overall time remain the same.
SOLUTION:
ANSWER: Sample answer: The more frequently interest is
a. 2, 4, 8, 16 compounded, the higher the account balance
b
. becomes.
ANSWER:
Sample answer: The more frequently interest is
compounded, the higher the account balance
becomes.
x −
x
ERROR ANALYSIS
c. y = 2 43. Beth and Liz are solving 6
x 3 −x − 1
d > 36 . Is either of them correct? Explain your
. y = 0.003(2) reasoning.
e. about 3,221,225.47 in.
42. WRITING IN MATHIn a problem about
compound interest, describe what happens as the
compounding period becomes more frequent while
the principal and overall time remain the same.
SOLUTION:
Sample answer: The more frequently interest is
compounded, the higher the account balance
becomes.
ANSWER:
Sample answer: The more frequently interest is
compounded, the higher the account balance
becomes.
x −
ERROR ANALYSIS
43. Beth and Liz are solving 6
3 > 36−x − 1. Is either of them correct? Explain your
reasoning.
SOLUTION:
Sample answer: Beth; Liz added the exponents
instead of multiplying them when taking the power of
a power.
ANSWER:
Sample answer: Beth; Liz added the exponents
instead of multiplying them when taking the power of
a power.
18 18 18
44. CHALLENGESolve for x: 16 + 16 + 16 +
18 18 x
16 + 16 = 4 .
SOLUTION:
SOLUTION:
Sample answer: Beth; Liz added the exponents
instead of multiplying them when taking the power of
a power.
ANSWER:
Sample answer: Beth; Liz added the exponents ANSWER:
instead of multiplying them when taking the power of 37.1610
a power.
45. OPEN ENDEDWhat would be a more beneficial
18 18 18 change to a 5-year loan at 8% interest compounded
44. CHALLENGESolve for x: 16 + 16 + 16 + monthly: reducing the term to 4 years or reducing the
18 18 x
16 + 16 = 4 . interest rate to 6.5%?
SOLUTION:
SOLUTION:
Reducing the term will be more beneficial. The
multiplier is 1.3756 for the 4-year and 1.3828 for the
6.5%.
ANSWER:
Reducing the term will be more beneficial. The
multiplier is 1.3756 for the 4-year and 1.3828 for the
6.5%.
CCSS ARGUMENTS
46. Determine whether the
following statements are sometimes, always, or
ANSWER: never true. Explain your reasoning.
37.1610 a. x 20x
2 > −8 for all values of x.
b. The graph of an exponential growth equation is
45. OPEN ENDEDWhat would be a more beneficial increasing.
change to a 5-year loan at 8% interest compounded
monthly: reducing the term to 4 years or reducing the c. The graph of an exponential decay equation is
interest rate to 6.5%? increasing.
SOLUTION:
SOLUTION:
Reducing the term will be more beneficial. The a. x 20x
multiplier is 1.3756 for the 4-year and 1.3828 for the Always; 2 will always be positive, and −8 will
always be negative.
6.5%. b. Always; by definition the graph will always be
increasing even if it is a small increase.
ANSWER:
Reducing the term will be more beneficial. The c. Never; by definition the graph will always be
multiplier is 1.3756 for the 4-year and 1.3828 for the decreasing even if it is a small decrease.
6.5%.
ANSWER:
a x 20x
. Always; 2 will always be positive, and 8 will
−
CCSS ARGUMENTS
46. Determine whether the always be negative.
following statements are sometimes, always, or b
never true. Explain your reasoning. . Always; by definition the graph will always be
a. x 20x increasing even if it is a small increase.
2 > −8 for all values of x. c. Never; by definition the graph will always be
b. The graph of an exponential growth equation is decreasing even if it is a small decrease.
increasing.
c. The graph of an exponential decay equation is
OPEN ENDEDWrite an exponential inequality with
increasing. 47.
a solution of x 2.
≤
SOLUTION:
x 20x SOLUTION:
a. Always; 2 will always be positive, and −8 will x 2
Sample answer: 4 4
always be negative. ≤
b. Always; by definition the graph will always be
increasing even if it is a small increase. ANSWER:
x 2
Sample answer: 4 4
≤
c. Never; by definition the graph will always be
decreasing even if it is a small decrease. 2x x + 1 2x + 2 4x + 1
PROOFShow that 27 · 81 = 3 · 9 .
48.
ANSWER:
x 20x SOLUTION:
a
. Always; 2 will always be positive, and 8 will
−
always be negative.
b
. Always; by definition the graph will always be
increasing even if it is a small increase.
c. Never; by definition the graph will always be
decreasing even if it is a small decrease.
OPEN ENDEDWrite an exponential inequality with
47.
a solution of x 2.
≤
SOLUTION:
x 2
Sample answer: 4 4
≤
ANSWER:
ANSWER:
x 2
Sample answer: 4 4
≤
2x x + 1 2x + 2 4x + 1
PROOFShow that 27 · 81 = 3 · 9 .
48.
SOLUTION:
WRITING IN MATHIf you were given the initial
49.
and final amounts of a radioactive substance and the
amount of time that passes, how would you
determine the rate at which the amount was
increasing or decreasing in order to write an
equation?
SOLUTION:
Sample answer: Divide the final amount by the initial
amount. If n is the number of time intervals that pass,
ANSWER: take the nth root of the answer.
ANSWER:
Sample answer: Divide the final amount by the initial
amount. If n is the number of time intervals that pass,
take the nth root of the answer.
−4 =
50. 3 × 10
A 30,000
−
B 0.0003
WRITING IN MATHIf you were given the initial
49. C 120
and final amounts of a radioactive substance and the −
amount of time that passes, how would you D 0.00003
determine the rate at which the amount was
increasing or decreasing in order to write an SOLUTION:
equation?
SOLUTION:
Sample answer: Divide the final amount by the initial
amount. If n is the number of time intervals that pass,
take the nth root of the answer.
ANSWER: B is the correct option.
Sample answer: Divide the final amount by the initial
amount. If n is the number of time intervals that pass, ANSWER:
B
take the nth root of the answer.
−4 = 51. Which of the following could not be a solution to 5 −
50. 3 × 10 3x < 3?
A 30,000 −
−
B 0.0003 F2.5
C 120 G3
− H 3.5
D 0.00003 J
4
SOLUTION:
SOLUTION:
Check the inequality by substituting 2.5 for x.
So, F is the correct option.
B is the correct option.
ANSWER:
ANSWER: F
B
GRIDDED RESPONSEThe three angles of a
Which of the following could not be a solution to 5 52.
51. − triangle are 3x, x + 10, and 2x 40. Find the measure
3x < 3? −
− of the smallest angle in the triangle.
F2.5
G3
SOLUTION:
H3.5 Sum of the three angles in a triangle is 180 .
º
J 4
SOLUTION:
Check the inequality by substituting 2.5 for x.
So, F is the correct option.
ANSWER:
F
The measure of the smallest angle in the triangle is
30 .
º
GRIDDED RESPONSEThe three angles of a
52.
triangle are 3x, x + 10, and 2x − 40. Find the measure
of the smallest angle in the triangle. ANSWER:
30
SOLUTION:
Sum of the three angles in a triangle is 180 . SAT/ACT Which of the following is equivalent to
º 53.
(x)(x)(x)(x) for all x?
A x + 4
B 4x
C 2x2
D 4x2
E 4
x
SOLUTION:
The measure of the smallest angle in the triangle is
30 .
º E is the correct choice.
ANSWER:
30 ANSWER:
E
SAT/ACT Which of the following is equivalent to
53. Graph each function.
(x)(x)(x)(x) for all x? x
y = 2(3)
54.
A x + 4
SOLUTION:
B 4x Make a table of values. Then plot the points and
C 2x2 sketch the graph.
D 4x2
E 4
x
SOLUTION:
E is the correct choice.
ANSWER:
E
Graph each function.
x
y = 2(3)
54.
SOLUTION:
Make a table of values. Then plot the points and
sketch the graph.
ANSWER:
x
y = 5(2)
55.
SOLUTION:
Make a table of values. Then plot the points and
sketch the graph.
ANSWER:
x
y = 5(2)
55.
SOLUTION:
Make a table of values. Then plot the points and
sketch the graph.
ANSWER:
56.
SOLUTION:
Make a table of values. Then plot the points and
sketch the graph.
ANSWER:
56.
SOLUTION:
Make a table of values. Then plot the points and
sketch the graph.
ANSWER:
Solve each equation.
57.
SOLUTION:
ANSWER:
ANSWER:
4
58.
SOLUTION:
Solve each equation.
57.
SOLUTION:
ANSWER:
18
59.
ANSWER:
4 SOLUTION:
58.
SOLUTION:
ANSWER:
8.5
60.
SOLUTION:
ANSWER:
18
59.
SOLUTION:
The square root of x cannot be negative, so there is
no solution.
ANSWER:
no solution
ANSWER:
8.5
61.
SOLUTION:
60.
SOLUTION:
ANSWER:
5
The square root of x cannot be negative, so there is
no solution.
62.
ANSWER:
SOLUTION:
no solution
61.
SOLUTION:
ANSWER:
20
−
ANSWER: 63.
5
SOLUTION:
62.
SOLUTION:
ANSWER:
5
ANSWER: 64.
20
−
SOLUTION:
63.
SOLUTION:
ANSWER:
ANSWER:
5
65.
64.
SOLUTION:
SOLUTION:
ANSWER:
1
−
ANSWER: SALES A salesperson earns $10 an hour plus a 10%
66.
commission on sales. Write a function to describe the
salesperson’s income. If the salesperson wants to
earn $1000 in a 40-hour week, what should his sales
be?
65.
SOLUTION:
SOLUTION: Let I be the income of the salesperson and m be his
sales.
The function that represent the situation is
.
Substitute 1000 for I in the equation and solve for m.
ANSWER:
1 ANSWER:
− I(m) = 400 + 0.1m; $6000
SALES A salesperson earns $10 an hour plus a 10%
66.
STATE FAIRA dairy makes three types of
commission on sales. Write a function to describe the 67.
cheese cheddar, Monterey Jack, and Swiss and
salesperson s income. If the salesperson wants to — —
’ sells the cheese in three booths at the state fair. At
earn $1000 in a 40-hour week, what should his sales the beginning of one day, the first booth received x
be? pounds of each type of cheese. The second booth
received y pounds of each type of cheese, and the
SOLUTION: third booth received z pounds of each type of cheese.
Let I be the income of the salesperson and m be his By the end of the day, the dairy had sold 131 pounds
sales. of cheddar, 291 pounds of Monterey Jack, and 232
The function that represent the situation is pounds of Swiss. The table below shows the percent
. of the cheese delivered in the morning that was sold
Substitute 1000 for I in the equation and solve for m. at each booth. How many pounds of cheddar cheese
did each booth receive in the morning?
ANSWER:
I(m) = 400 + 0.1m; $6000
STATE FAIRA dairy makes three types of
67.
cheese cheddar, Monterey Jack, and Swiss and SOLUTION:
— — The system of equations that represent the situation:
sells the cheese in three booths at the state fair. At
the beginning of one day, the first booth received x
pounds of each type of cheese. The second booth
received y pounds of each type of cheese, and the
third booth received z pounds of each type of cheese.
By the end of the day, the dairy had sold 131 pounds
of cheddar, 291 pounds of Monterey Jack, and 232
pounds of Swiss. The table below shows the percent
of the cheese delivered in the morning that was sold Eliminate the variable x by using two pairs of
equations.
at each booth. How many pounds of cheddar cheese
did each booth receive in the morning?
Subtract (1) and (2).
Multiply (2) by 3 and (3) by 4 and subtract both the
SOLUTION: equations.
The system of equations that represent the situation:
Solve the system of two equations:
Eliminate the variable x by using two pairs of
equations.
Subtract (1) and (2).
Substitute z = 100 in the equation
Multiply (2) by 3 and (3) by 4 and subtract both the
equations.
Substitute y = 150 and z = 100 in the (1) and solve
for x.
Solve the system of two equations:
Booth 1 has 190 lb; Booth 2 has 150 lb; Booth 3 has
100 lb.
Substitute z = 100 in the equation
ANSWER:
booth 1, 190 lb; booth 2, 150 lb; booth 3, 100 lb
Find [g h](x) and [h g](x).
◦ ◦
68. h(x) = 2x − 1
g(x) = 3x + 4
Substitute y = 150 and z = 100 in the (1) and solve SOLUTION:
for x.
Booth 1 has 190 lb; Booth 2 has 150 lb; Booth 3 has ANSWER:
100 lb. 6x + 1; 6x + 7
ANSWER: 2
booth 1, 190 lb; booth 2, 150 lb; booth 3, 100 lb 69. h(x) = x + 2
g(x) = x − 3
Find [g h](x) and [h g](x).
◦ ◦
h(x) = 2x 1 SOLUTION:
68. −
g(x) = 3x + 4
SOLUTION:
ANSWER:
x2 1; x2 6x + 11
− −
2
ANSWER: 70. h(x) = x + 1
g(x) = 2x + 1
6x + 1; 6x + 7 −
SOLUTION:
h(x) = x2 + 2
69.
g(x) = x − 3
SOLUTION:
ANSWER:
2x2 1; 4x2 4x + 2
− − −
h(x) = 5x
71. −
ANSWER:
2 2 g(x) = 3x − 5
x 1; x 6x + 11
− −
SOLUTION:
h(x) = x2 + 1
70.
g(x) = 2x + 1
−
SOLUTION:
ANSWER:
15x 5; 15x + 25
− − −
h(x) = x3
72.
g(x) = x − 2
SOLUTION:
ANSWER:
2x2 1; 4x2 4x + 2
− − −
h(x) = 5x
71. −
g(x) = 3x − 5
SOLUTION:
ANSWER:
3 3 2
x x 6x + 12x 8
−2; − −
73. h(x) = x + 4
g(x) = | x |
SOLUTION:
ANSWER:
15x 5; 15x + 25
− − −
h(x) = x3
72.
ANSWER:
g(x) = x − 2 | x + 4 | ; | x | + 4
SOLUTION:
ANSWER:
3 3 2
x x 6x + 12x 8
−2; − −
73. h(x) = x + 4
g(x) = | x |
SOLUTION:
ANSWER:
| x + 4 | ; | x | + 4
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