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Solve each equation.
5x 2x − 4
1. 3 = 27
SOLUTION:
Use the Property of Equality for Exponential
Functions.
2y − 3 y + 1
2. 16 = 4
SOLUTION:
Use the Property of Equality for Exponential
Functions.
6x x − 2
3. 2 = 32
SOLUTION:
Use the Property of Equality for Exponential
Functions.
7-2 Solving Exponential Equations and Inequalities
x + 5 8x − 6
Solve each equation. 4. 49 = 7
5x 2x − 4
1. 3 = 27
SOLUTION:
SOLUTION:
Use the Property of Equality for Exponential
Use the Property of Equality for Exponential Functions.
Functions.
2y − 3 y + 1
2. 16 = 4
SCIENCE
5. Mitosis is a process in which one cell
SOLUTION: 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
Use the Property of Equality for Exponential many cells will there be in one hour?
Functions.
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.
6x x − 2
3. 2 = 32
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.
Substitute $500 for P, 0.0225 for r, 26 for n and 6 for
Use the Property of Equality for Exponential t and simplify.
Functions.
x + 5 8x − 6
4. 49 = 7
SOLUTION:
Solve each inequality.
eSolutions Manual - Powered by Cognero 2x + 6 2x – 4 Page1
7. 4 ≤64
SOLUTION:
Use the Property of Equality for Exponential
Functions.
Use the Property of Inequality 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. 8.
b. If you begin with one Escherichia coli cell, how
many cells will there be in one hour? SOLUTION:
SOLUTION:
a.
The exponential function that represent the number
of cells after t minutes is .
b. Use the Property of Inequality for Exponential
Substitute 1 for t in the function and solve for c. Functions.
6. A certificate of deposit (CD) pays 2.25% annual
interest compounded biweekly. If you deposit $500 Solve each equation.
into this CD, what will the balance be after 6 years? 4x + 2
9. 8 = 64
SOLUTION:
Use the compound interest formula. SOLUTION:
Substitute $500 for P, 0.0225 for r, 26 for n and 6 for
t and simplify.
Use the Property of Equality for Exponential
Functions.
Solve each inequality. x − 6
2x + 6 2x – 4 10. 5 = 125
7. 4 ≤64
SOLUTION:
SOLUTION:
Use the Property of Equality for Exponential
Functions.
Use the Property of Inequality for Exponential
Functions.
a + 2 3a + 1
11. 81 = 3
SOLUTION:
8.
SOLUTION: Use the Property of Equality for Exponential
Functions.
Use the Property of Inequality for Exponential b + 2 2 − 2b
Functions. 12. 256 = 4
SOLUTION:
Solve each equation.
4x + 2
9. 8 = 64
Use the Property of Equality for Exponential
SOLUTION: Functions.
Use the Property of Equality for Exponential
Functions.
3c + 1 3c − 1
13. 9 = 27
SOLUTION:
x − 6
10. 5 = 125
SOLUTION:
Use the Property of Equality for Exponential
Functions.
Use the Property of Equality for Exponential
Functions.
2y + 4 y + 1
a + 2 3a + 1 14. 8 = 16
11. 81 = 3
SOLUTION:
SOLUTION:
Use the Property of Equality for Exponential Use the Property of Equality for Exponential
Functions. Functions.
b + 2 2 − 2b
12. 256 = 4
15. CCSS MODELINGIn 2009, My-Lien received
SOLUTION: $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.
Use the Property of Equality for Exponential b. Assume that the amount of money continues to
Functions. 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
exponential function and simplify.
3c + 1 3c − 1
13. 9 = 27
SOLUTION:
Use the Property of Equality for Exponential The exponential function that models the situation
Functions. is .
b.
Substitute 22 for x in the modeled function and solve
for y.
2y + 4 y + 1
14. 8 = 16
SOLUTION: 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.
Use the Property of Equality for Exponential
Functions.
15. CCSS MODELINGIn 2009, My-Lien received
$10,000 from her grandmother. Her parents invested An exponential function that passes through the given
all of the money, and by 2021, the amount will have points is .
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, 17. (0, 256) and (4, 81)
the number of years since 2009.
b. Assume that the amount of money continues to SOLUTION:
grow at the same rate. What would be the balance in Substitute 81 for y and 256 for a and 4 for x into an
the account in 2031? exponential function and determine the value of b.
SOLUTION:
a.
Substitute 16780 for y 10000 for a and 12 for x in the
exponential function and simplify.
An exponential function that passes through the given
points is .
18. (0, 128) and (5, 371,293)
The exponential function that models the situation
is . SOLUTION:
Substitute 371293 for y and 128 for a and 5 for x into
b. an exponential function and determine the value of b.
Substitute 22 for x in the modeled function and solve
for y.
Write an exponential function for the graph that
passes through the given points.
16. (0, 6.4) and (3, 100) An exponential function that passes through the given
points is .
SOLUTION:
Substitute 100 for y and 6.4 for a and 3 for x into an 19. (0, 144), and (4, 21,609)
exponential function and determine the value of b.
SOLUTION:
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 .
17. (0, 256) and (4, 81) An exponential function that passes through the given
points is .
SOLUTION:
Substitute 81 for y and 256 for a and 4 for x into an 20. Find the balance of an account after 7 years if $700
exponential function and determine the value of b. is deposited into an account paying 4.3% interest
compounded monthly.
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 .
18. (0, 128) and (5, 371,293)
SOLUTION: 21. Determine how much is in a retirement account after
Substitute 371293 for y and 128 for a and 5 for x into 20 years if $5000 was invested at 6.05% interest
an exponential function and determine the value of b. compounded weekly.
SOLUTION:
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 .
19. (0, 144), and (4, 21,609)
22. A savings account offers 0.7% interest compounded
SOLUTION: bimonthly. If $110 is deposited in this account, what
Substitute 21609 for y and 144 for a and 4 for x into will the balance be after 15 years?
an exponential function and determine the value of b.
SOLUTION:
Use the compound interest formula.
Substitute $110 for P, 0.007 for r, 6 for n and 15 for t
and simplify.
An exponential function that passes through the given
points is .
20. Find the balance of an account after 7 years if $700 23. A college savings account pays 13.2% annual
is deposited into an account paying 4.3% interest interest compounded semiannually. What is the
compounded monthly. balance of an account after 12 years if $21,000 was
initially deposited?
SOLUTION:
Use the compound interest formula. SOLUTION:
Substitute $700 for P, 0.043 for r, 12 for n and 7 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.
21. Determine how much is in a retirement account after
20 years if $5000 was invested at 6.05% interest Solve each inequality.
compounded weekly.
24.
SOLUTION:
Use the compound interest formula. SOLUTION:
Substitute $5000 for P, 0.0605 for r, 52 for n and 20
for t and simplify.
Use the Property of Inequality for Exponential
Functions.
22. A savings account offers 0.7% interest compounded 25.
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.
23. A college savings account pays 13.2% annual 26.
interest compounded semiannually. What is the
balance of an account after 12 years if $21,000 was SOLUTION:
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.
Use the Property of Inequality for Exponential
Functions.
Solve each inequality.
24.
SOLUTION:
27.
SOLUTION:
Use the Property of Inequality for Exponential
Functions.
Use the Property of Inequality for Exponential
25. Functions.
SOLUTION:
Use the Property of Inequality for Exponential
Functions.
28.
SOLUTION:
26.
SOLUTION:
Use the Property of Inequality for Exponential
Functions.
Use the Property of Inequality for Exponential 29.
Functions.
SOLUTION:
27.
Use the Property of Inequality for Exponential
SOLUTION: Functions.
Use the Property of Inequality for Exponential
Functions. 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
minutes.
28.
c.
The optimum drinking temperature is 60°C. Will
the mug of hot chocolate be at or below this
SOLUTION: temperature after 10 minutes?
SOLUTION:
a.
Substitute 15 for t in the equation and simplify.
b.
Use the Property of Inequality for Exponential Substitute 30 for t in the equation and simplify.
Functions.
c.
Substitute 10 for t in the equation and simplify.
So, temperature of the hot chocolate will be below
29.
60°C after 10 minutes.
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 Inequality 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.
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 The equation for the area a defended by a beaver
minutes. weighting w pounds is
b. Find the temperature of the hot chocolate after 30 b.
minutes. Substitute 430 for w in the equation and solve for y.
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. Solve each equation.
Substitute 15 for t in the equation and simplify.
32.
b. SOLUTION:
Substitute 30 for t in the equation and simplify.
c.
Substitute 10 for t in the equation and simplify.
Use the Property of Equality for Exponential
Functions.
So, temperature of the hot chocolate will be below
60°C after 10 minutes.
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. 33.
b. Scientists believe that thousands of years ago, the
beaver’s ancestors were 11 feet long and weighed SOLUTION:
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.
Use the Property of Equality for Exponential
Functions.
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. 34.
SOLUTION:
Solve each equation.
32.
Use the Property of Equality for Exponential
SOLUTION: Functions.
35.
Use the Property of Equality for Exponential
Functions.
SOLUTION:
33. Use the Property of Equality for Exponential
Functions.
SOLUTION:
Use the Property of Equality for Exponential 36.
Functions.
SOLUTION:
34.
SOLUTION: Use the Property of Equality for Exponential
Functions.
Use the Property of Equality for Exponential
Functions.
37.
SOLUTION:
35.
SOLUTION:
Use the Property of Equality for Exponential
Functions.
Use the Property of Equality for Exponential
Functions.
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
value of b to the nearest ten-thousandth.)
36. b. Suppose the population continued to grow at that
rate. Estimate the population in 2000.
SOLUTION:
c. In 2000, the population of the world was about
6.08 billion. Compare your estimate to the actual
population.
d. a
Use the equation you wrote in part to estimate
the world population in the year 2020. How accurate
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
exponential function and solve for b.
Use the Property of Equality for Exponential
Functions.
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
even higher than the actual population will be at the
38. CCSS MODELINGIn 1950, the world population time.
was about 2.556 billion. By 1980, it had increased to
about 4.458 billion. 39. TREES The diameter of the base of a tree trunk in
a. x
Write an exponential function of the form y = ab
that could be used to model the world population y in centimeters varies directly with the power of its
billions for 1950 to 1980. Write the equation in terms height in meters.
of x, the number of years since 1950. (Round the a. A young sequoia tree is 6 meters tall, and the
value of b to the nearest ten-thousandth.)
b. Suppose the population continued to grow at that diameter of its base is 19.1 centimeters. Use this
rate. Estimate the population in 2000. information to write an equation for the diameter d of
the base of a sequoia tree if its height is h meters
c. In 2000, the population of the world was about
6.08 billion. Compare your estimate to the actual high
population. b. The General Sherman Tree in Sequoia National
d. a Park, California, is approximately 84 meters tall.
Use the equation you wrote in part to estimate Find the diameter of the General Sherman Tree at its
the world population in the year 2020. How accurate
do you think the estimate is? Explain your reasoning. base.
SOLUTION:
a. SOLUTION:
Substitute 4.458 for y, 2.556 for a, and 30 for x in the a.
exponential function and solve for b. The equation that represent the situation is
.
b.
Substitute 84 for h in the equation and solve for d.
The exponential function that model the situation is
.
b. The diameter of the General Sherman Tree at its
Substitute 50 for x in the equation and simplify. base is 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
c. The prediction was about 375 million greater than the minimum deposits.
the actual population. b. Draw a graph to show the balances for each
d.
investment option after t years.
Substitute 70 for x in the equation and simplify.
c. Explain whether Option A or Option B is the
better investment choice.
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
time.
39. TREES The diameter of the base of a tree trunk in
centimeters varies directly with the power of its SOLUTION:
height in meters. a.
Use the compound interest formula.
a. A young sequoia tree is 6 meters tall, and the The equation that represents Option A
diameter of its base is 19.1 centimeters. Use this
information to write an equation for the diameter d of is .
the base of a sequoia tree if its height is h meters The equation that represents Option B
high
b. The General Sherman Tree in Sequoia National is
Park, California, is approximately 84 meters tall.
Find the diameter of the General Sherman Tree at its b.
base. The graph that shows the balances for each
investment option after t years:
SOLUTION:
a.
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.
c.
40. FINANCIAL LITERACYMrs. Jackson has two During the first 22 years, Option B is the better
different retirement investment plans from which to choice because the total is greater than that of
choose. Option A. However, after about 22 years, the
a. Write equations for Option A and Option B given balance of Option A exceeds that of Option B, so
the minimum deposits. Option A is the better choice.
b. Draw a graph to show the balances for each
investment option after t years.
41. MULTIPLE REPRESENTATIONSIn this
c. Explain whether Option A or Option B is the problem, you will explore the rapid increase of an
better investment choice. 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.
SOLUTION:
c. SYMBOLIC
Use the pattern in the table to write
a. an equation for the number of pieces in the stack
Use the compound interest formula.
after x cuts.
The equation that represents Option A d. ANALYTICAL
The thickness of ordinary paper
is . is about 0.003 inch. Write an equation for the
thickness of the stack of paper after x cuts.
e.ANALYTICAL
The equation that represents Option B How thick will the stack of
paper be after 30 cuts?
is
SOLUTION:
b. a.
The graph that shows the balances for each There will be 2, 4, 8, 16 pieces after the first, second,
investment option after t years: third and fourth cut respectively.
b.
c.
The equation that represent the situation is
d.
Substitute 0.003 for a and 2 for b in the exponential
function.
e.
c. Substitute 30 for x in the equation
During the first 22 years, Option B is the better and
choice because the total is greater than that of simplify.
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 The thickness of the stack of paper after 30 cuts is
problem, you will explore the rapid increase of an about 3221225.47 in.
exponential function. A large sheet of paper is cut in
half, and one of the resulting pieces is placed on top
WRITING IN MATHIn a problem about
of the other. Then the pieces in the stack are cut in 42.
half and placed on top of each other. Suppose this compound interest, describe what happens as the
procedure is repeated several times. compounding period becomes more frequent while
a. CONCRETE the principal and overall time remain the same.
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 SOLUTION:
many pieces after the third cut? How many pieces Sample answer: The more frequently interest is
after the fourth cut? compounded, the higher the account balance
b. TABULAR becomes.
Record your results in a table.
x −
c. SYMBOLIC
Use the pattern in the table to write
ERROR ANALYSIS
an equation for the number of pieces in the stack 43. Beth and Liz are solving 6
3 > 36−x − 1. Is either of them correct? Explain your
after x cuts. reasoning.
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?
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.
Substitute 0.003 for a and 2 for b in the exponential
function.
e.
Substitute 30 for x in the equation and
simplify. SOLUTION:
Sample answer: Beth; Liz added the exponents
instead of multiplying them when taking the power of
a power.
18 18 18
CHALLENGESolve for x: 16 + 16 + 16 +
44.
18 18 x
The thickness of the stack of paper after 30 cuts is 16 + 16 = 4 .
about 3221225.47 in.
SOLUTION:
WRITING IN MATHIn a problem about Simplify the exponential equation.
42.
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. Use a graphing calculator to solve the exponential
x − equation for x.
ERROR ANALYSIS
43. Beth and Liz are solving 6 Enter 4x as Y1 and 2.36118 E22 as Y2. Adjust the
3 > 36−x − 1. Is either of them correct? Explain your viewing window to see both graphs and the
reasoning. intersection.
Use the CALC function to find the intersection of
the two graphs.
So, the solution to the equation is about x =
37.1610.
OPEN ENDEDWhat would be a more beneficial
45.
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%.
SOLUTION:
CCSS ARGUMENTS
Sample answer: Beth; Liz added the exponents 46. Determine whether the
instead of multiplying them when taking the power of following statements are sometimes, always, or
never true. Explain your reasoning.
a power. a. x 20x
2 > 8 for all values of x.
−
18 18 18 b. The graph of an exponential growth equation is
CHALLENGESolve for x: 16 + 16 + 16 +
44. increasing.
18 18 x
16 + 16 = 4 .
c. The graph of an exponential decay equation is
increasing.
SOLUTION:
Simplify the exponential equation.
SOLUTION:
x 20x
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.
Use a graphing calculator to solve the exponential
OPEN ENDEDWrite an exponential inequality with
equation for x. 47.
a solution of x 2.
Enter 4x as Y1 and 2.36118 E22 as Y2. Adjust the ≤
viewing window to see both graphs and the
intersection. SOLUTION:
x 2
Sample answer: 4 4
Use the CALC function to find the intersection of ≤
the two graphs. 2x x + 1 2x + 2 4x + 1
PROOFShow that 27 · 81 = 3 · 9 .
48.
SOLUTION:
So, the solution to the equation is about x =
37.1610.
WRITING IN MATHIf you were given the initial
49.
OPEN ENDEDWhat would be a more beneficial
45. and final amounts of a radioactive substance and the
change to a 5-year loan at 8% interest compounded amount of time that passes, how would you
monthly: reducing the term to 4 years or reducing the determine the rate at which the amount was
interest rate to 6.5%? increasing or decreasing in order to write an
equation?
SOLUTION:
Reducing the term will be more beneficial. The
multiplier is 1.3756 for the 4-year and 1.3828 for the SOLUTION:
Sample answer: Divide the final amount by the initial
6.5%. amount. If n is the number of time intervals that pass,
take the nth root of the answer.
CCSS ARGUMENTS
46. Determine whether the
following statements are sometimes, always, or −4 =
never true. Explain your reasoning. 50. 3 × 10
x 20x A 30,000
a. 2 > 8 for all values of x. −
− B 0.0003
b. The graph of an exponential growth equation is C 120
increasing. −
D 0.00003
c. The graph of an exponential decay equation is
increasing.
SOLUTION:
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. B is the correct option.
51. Which of the following could not be a solution to 5 −
OPEN ENDEDWrite an exponential inequality with
47. 3x < 3?
a solution of x 2. −
≤
F2.5
SOLUTION:
x 2 G3
Sample answer: 4 4 H
≤ 3.5
J
2x x + 1 2x + 2 4x + 1 4
PROOFShow that 27 · 81 = 3 · 9 .
48.
SOLUTION:
SOLUTION: Check the inequality by substituting 2.5 for x.
So, F is the correct option.
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
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,
take the nth root of the answer.
−4 =
50. 3 × 10
A 30,000
− The measure of the smallest angle in the triangle is
B 0.0003 30 .
º
C 120
− SAT/ACT
D 0.00003 53. Which of the following is equivalent to
(x)(x)(x)(x) for all x?
SOLUTION:
A x + 4
B 4x
C 2x2
D 4x2
E 4
x
B is the correct option.
SOLUTION:
51. Which of the following could not be a solution to 5 −
3x < 3?
−
F2.5
G3 E is the correct choice.
H 3.5 Graph each function.
J
4 x
y = 2(3)
54.
SOLUTION:
Check the inequality by substituting 2.5 for x. SOLUTION:
Make a table of values. Then plot the points and
sketch the graph.
So, F is the correct option.
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
x
y = 5(2)
The measure of the smallest angle in the triangle is 55.
30 .
º
SOLUTION:
SAT/ACT Which of the following is equivalent to Make a table of values. Then plot the points and
53. sketch the graph.
(x)(x)(x)(x) for all x?
A x + 4
B 4x
C 2x2
D 4x2
E 4
x
SOLUTION:
E is the correct choice.
Graph each function.
x
y = 2(3)
54.
SOLUTION:
Make a table of values. Then plot the points and
sketch the graph.
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.
Solve each equation.
57.
SOLUTION:
58.
SOLUTION:
56.
59.
SOLUTION:
Make a table of values. Then plot the points and SOLUTION:
sketch the graph.
60.
SOLUTION:
The square root of x cannot be negative, so there is
no solution.
Solve each equation. 61.
SOLUTION:
57.
SOLUTION:
62.
58.
SOLUTION:
SOLUTION:
59.
63.
SOLUTION:
SOLUTION:
60.
SOLUTION:
64.
SOLUTION:
The square root of x cannot be negative, so there is
no solution.
61.
SOLUTION:
65.
SOLUTION:
62.
SOLUTION:
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:
Let I be the income of the salesperson and m be his
sales.
The function that represent the situation is
63. .
Substitute 1000 for I in the equation and solve for m.
SOLUTION:
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
64. third booth received z pounds of each type of cheese.
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?
65.
SOLUTION:
The system of equations that represent the situation:
SOLUTION:
Eliminate the variable x by using two pairs of
equations.
SALES A salesperson earns $10 an hour plus a 10% Subtract (1) and (2).
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
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:
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.
By the end of the day, the dairy had sold 131 pounds Substitute z = 100 in the equation
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.
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.
h(x) = x2 + 2
69.
g(x) = x − 3
SOLUTION:
Solve the system of two equations:
h(x) = x2 + 1
70.
g(x) = 2x + 1
Substitute z = 100 in the equation −
SOLUTION:
Substitute y = 150 and z = 100 in the (1) and solve
for x.
h(x) = 5x
71. −
g(x) = 3x − 5
SOLUTION:
Booth 1 has 190 lb; Booth 2 has 150 lb; Booth 3 has
100 lb.
h(x) = x3
Find [g h](x) and [h g](x). 72.
◦ ◦
h(x) = 2x 1 g(x) = x − 2
68. −
g(x) = 3x + 4
SOLUTION:
SOLUTION:
73. h(x) = x + 4
g(x) = | x |
SOLUTION:
h(x) = x2 + 2
69.
g(x) = x − 3
SOLUTION:
h(x) = x2 + 1
70.
g(x) = 2x + 1
−
SOLUTION:
h(x) = 5x
71. −
g(x) = 3x − 5
SOLUTION:
h(x) = x3
72.
g(x) = x − 2
SOLUTION:
73. h(x) = x + 4
g(x) = | x |
SOLUTION:
Solve each equation.
5x 2x − 4
1. 3 = 27
SOLUTION:
Use the Property of Equality for Exponential
Functions.
2y − 3 y + 1
2. 16 = 4
SOLUTION:
Use the Property of Equality for Exponential
Functions.
6x x − 2
3. 2 = 32
SOLUTION:
Use the Property of Equality for Exponential
Functions.
x + 5 8x − 6
Solve each equation. 4. 49 = 7
5x 2x − 4
1. 3 = 27
SOLUTION:
SOLUTION:
Use the Property of Equality for Exponential
Use the Property of Equality for Exponential Functions.
Functions.
2y − 3 y + 1
2. 16 = 4 SCIENCE
5. Mitosis is a process in which one cell
SOLUTION: 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
Use the Property of Equality for Exponential many cells will there be in one hour?
Functions. 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.
6x x − 2
3. 2 = 32
6. A certificate of deposit (CD) pays 2.25% annual
SOLUTION: 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
Use the Property of Equality for Exponential t and simplify.
Functions.
x + 5 8x − 6
4. 49 = 7
SOLUTION:
Solve each inequality.
2x + 6 2x – 4
7. 4 ≤64
SOLUTION:
Use the Property of Equality for Exponential
Functions.
Use the Property of Inequality 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. 8.
b. If you begin with one Escherichia coli cell, how
many cells will there be in one hour? SOLUTION:
SOLUTION:
a.
The exponential function that represent the number
of cells after t minutes is .
b. Use the Property of Inequality for Exponential
Substitute 1 for t in the function and solve for c. Functions.
6. A certificate of deposit (CD) pays 2.25% annual
interest compounded biweekly. If you deposit $500 Solve each equation.
into this CD, what will the balance be after 6 years? 4x + 2
9. 8 = 64
SOLUTION:
Use the compound interest formula. SOLUTION:
Substitute $500 for P, 0.0225 for r, 26 for n and 6 for
t and simplify.
Use the Property of Equality for Exponential
Functions.
7-2 Solving Exponential Equations and Inequalities
Solve each inequality. x − 6
2x + 6 2x – 4 10. 5 = 125
7. 4 ≤64
SOLUTION:
SOLUTION:
Use the Property of Equality for Exponential
Functions.
Use the Property of Inequality for Exponential
Functions.
a + 2 3a + 1
11. 81 = 3
SOLUTION:
8.
SOLUTION: Use the Property of Equality for Exponential
Functions.
Use the Property of Inequality for Exponential b + 2 2 − 2b
Functions. 12. 256 = 4
SOLUTION:
Solve each equation.
4x + 2
9. 8 = 64
Use the Property of Equality for Exponential
SOLUTION: Functions.
Use the Property of Equality for Exponential
Functions.
3c + 1 3c − 1
13. 9 = 27
SOLUTION:
x − 6
10. 5 = 125
SOLUTION:
Use the Property of Equality for Exponential
eSolutions Manual - Powered by Cognero Functions. Page2
Use the Property of Equality for Exponential
Functions.
2y + 4 y + 1
a + 2 3a + 1 14. 8 = 16
11. 81 = 3
SOLUTION:
SOLUTION:
Use the Property of Equality for Exponential Use the Property of Equality for Exponential
Functions. Functions.
b + 2 2 − 2b
12. 256 = 4
15. CCSS MODELINGIn 2009, My-Lien received
SOLUTION: $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.
Use the Property of Equality for Exponential b. Assume that the amount of money continues to
Functions. 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
exponential function and simplify.
3c + 1 3c − 1
13. 9 = 27
SOLUTION:
Use the Property of Equality for Exponential The exponential function that models the situation
Functions. is .
b.
Substitute 22 for x in the modeled function and solve
for y.
2y + 4 y + 1
14. 8 = 16
SOLUTION: 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.
Use the Property of Equality for Exponential
Functions.
15. CCSS MODELINGIn 2009, My-Lien received
$10,000 from her grandmother. Her parents invested An exponential function that passes through the given
all of the money, and by 2021, the amount will have points is .
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, 17. (0, 256) and (4, 81)
the number of years since 2009.
b. Assume that the amount of money continues to SOLUTION:
grow at the same rate. What would be the balance in Substitute 81 for y and 256 for a and 4 for x into an
the account in 2031? exponential function and determine the value of b.
SOLUTION:
a.
Substitute 16780 for y 10000 for a and 12 for x in the
exponential function and simplify.
An exponential function that passes through the given
points is .
18. (0, 128) and (5, 371,293)
The exponential function that models the situation
is . SOLUTION:
Substitute 371293 for y and 128 for a and 5 for x into
b. an exponential function and determine the value of b.
Substitute 22 for x in the modeled function and solve
for y.
Write an exponential function for the graph that
passes through the given points. An exponential function that passes through the given
16. (0, 6.4) and (3, 100)
points is .
SOLUTION:
Substitute 100 for y and 6.4 for a and 3 for x into an 19. (0, 144), and (4, 21,609)
exponential function and determine the value of b.
SOLUTION:
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 .
17. (0, 256) and (4, 81) An exponential function that passes through the given
points is .
SOLUTION:
Substitute 81 for y and 256 for a and 4 for x into an 20. Find the balance of an account after 7 years if $700
exponential function and determine the value of b. is deposited into an account paying 4.3% interest
compounded monthly.
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 .
18. (0, 128) and (5, 371,293)
SOLUTION: 21. Determine how much is in a retirement account after
Substitute 371293 for y and 128 for a and 5 for x into 20 years if $5000 was invested at 6.05% interest
an exponential function and determine the value of b. compounded weekly.
SOLUTION:
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 .
19. (0, 144), and (4, 21,609) 22. A savings account offers 0.7% interest compounded
SOLUTION: bimonthly. If $110 is deposited in this account, what
Substitute 21609 for y and 144 for a and 4 for x into will the balance be after 15 years?
an exponential function and determine the value of b.
SOLUTION:
Use the compound interest formula.
Substitute $110 for P, 0.007 for r, 6 for n and 15 for t
and simplify.
An exponential function that passes through the given
points is .
20. Find the balance of an account after 7 years if $700 23. A college savings account pays 13.2% annual
is deposited into an account paying 4.3% interest interest compounded semiannually. What is the
compounded monthly. balance of an account after 12 years if $21,000 was
initially deposited?
SOLUTION:
Use the compound interest formula. SOLUTION:
Substitute $700 for P, 0.043 for r, 12 for n and 7 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.
21. Determine how much is in a retirement account after
20 years if $5000 was invested at 6.05% interest Solve each inequality.
compounded weekly.
24.
SOLUTION:
SOLUTION:
Use the compound interest formula.
Substitute $5000 for P, 0.0605 for r, 52 for n and 20
for t and simplify.
Use the Property of Inequality for Exponential
Functions.
22. A savings account offers 0.7% interest compounded 25.
bimonthly. If $110 is deposited in this account, what
SOLUTION:
will the balance be after 15 years?
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.
23. A college savings account pays 13.2% annual 26.
interest compounded semiannually. What is the
balance of an account after 12 years if $21,000 was SOLUTION:
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.
Use the Property of Inequality for Exponential
Functions.
Solve each inequality.
24.
SOLUTION:
27.
SOLUTION:
Use the Property of Inequality for Exponential
Functions.
Use the Property of Inequality for Exponential
25. Functions.
SOLUTION:
Use the Property of Inequality for Exponential
Functions.
28.
SOLUTION:
26.
SOLUTION:
Use the Property of Inequality for Exponential
Functions.
Use the Property of Inequality for Exponential 29.
Functions.
SOLUTION:
27.
Use the Property of Inequality for Exponential
SOLUTION: Functions.
Use the Property of Inequality for Exponential
Functions. 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
minutes.
28.
c.
The optimum drinking temperature is 60°C. Will
the mug of hot chocolate be at or below this
SOLUTION: temperature after 10 minutes?
SOLUTION:
a.
Substitute 15 for t in the equation and simplify.
b.
Use the Property of Inequality for Exponential Substitute 30 for t in the equation and simplify.
Functions.
c.
Substitute 10 for t in the equation and simplify.
So, temperature of the hot chocolate will be below
29.
60°C after 10 minutes.
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 Inequality 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.
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 The equation for the area a defended by a beaver
minutes. weighting w pounds is
b. Find the temperature of the hot chocolate after 30 b.
minutes. Substitute 430 for w in the equation and solve for y.
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. Solve each equation.
Substitute 15 for t in the equation and simplify.
32.
b. SOLUTION:
Substitute 30 for t in the equation and simplify.
c.
Substitute 10 for t in the equation and simplify.
Use the Property of Equality for Exponential
Functions.
So, temperature of the hot chocolate will be below
60°C after 10 minutes.
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. 33.
b. Scientists believe that thousands of years ago, the
beaver’s ancestors were 11 feet long and weighed SOLUTION:
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. Use the Property of Equality for Exponential
Functions.
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. 34.
SOLUTION:
Solve each equation.
32.
Use the Property of Equality for Exponential
SOLUTION: Functions.
35.
Use the Property of Equality for Exponential
Functions. SOLUTION:
33. Use the Property of Equality for Exponential
Functions.
SOLUTION:
Use the Property of Equality for Exponential 36.
Functions.
SOLUTION:
34.
SOLUTION: Use the Property of Equality for Exponential
Functions.
Use the Property of Equality for Exponential
Functions.
37.
SOLUTION:
35.
SOLUTION:
Use the Property of Equality for Exponential
Functions.
Use the Property of Equality for Exponential
Functions.
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
value of b to the nearest ten-thousandth.)
36. b. Suppose the population continued to grow at that
rate. Estimate the population in 2000.
SOLUTION: c. In 2000, the population of the world was about
6.08 billion. Compare your estimate to the actual
population.
d. a
Use the equation you wrote in part to estimate
the world population in the year 2020. How accurate
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
exponential function and solve for b.
Use the Property of Equality for Exponential
Functions.
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
even higher than the actual population will be at the
38. CCSS MODELINGIn 1950, the world population time.
was about 2.556 billion. By 1980, it had increased to
about 4.458 billion. 39. TREES The diameter of the base of a tree trunk in
a. x
Write an exponential function of the form y = ab
that could be used to model the world population y in centimeters varies directly with the power of its
billions for 1950 to 1980. Write the equation in terms height in meters.
of x, the number of years since 1950. (Round the a. A young sequoia tree is 6 meters tall, and the
value of b to the nearest ten-thousandth.) diameter of its base is 19.1 centimeters. Use this
b. Suppose the population continued to grow at that information to write an equation for the diameter d of
rate. Estimate the population in 2000. the base of a sequoia tree if its height is h meters
c. In 2000, the population of the world was about
6.08 billion. Compare your estimate to the actual high
population. b. The General Sherman Tree in Sequoia National
d. a Park, California, is approximately 84 meters tall.
Use the equation you wrote in part to estimate Find the diameter of the General Sherman Tree at its
the world population in the year 2020. How accurate
do you think the estimate is? Explain your reasoning. base.
SOLUTION:
a. SOLUTION:
Substitute 4.458 for y, 2.556 for a, and 30 for x in the a.
exponential function and solve for b. The equation that represent the situation is
.
b.
Substitute 84 for h in the equation and solve for d.
The exponential function that model the situation is
.
b. The diameter of the General Sherman Tree at its
Substitute 50 for x in the equation and simplify. base is 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.
c. The prediction was about 375 million greater than
the actual population. b. Draw a graph to show the balances for each
d. investment option after t years.
Substitute 70 for x in the equation and simplify. c. Explain whether Option A or Option B is the
better investment choice.
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
time.
39. TREES The diameter of the base of a tree trunk in
centimeters varies directly with the power of its SOLUTION:
a.
height in meters. Use the compound interest formula.
a. A young sequoia tree is 6 meters tall, and the The equation that represents Option A
diameter of its base is 19.1 centimeters. Use this is .
information to write an equation for the diameter d of
the base of a sequoia tree if its height is h meters The equation that represents Option B
high
b. The General Sherman Tree in Sequoia National is
Park, California, is approximately 84 meters tall.
Find the diameter of the General Sherman Tree at its b.
base. The graph that shows the balances for each
investment option after t years:
SOLUTION:
a.
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.
c.
40. FINANCIAL LITERACYMrs. Jackson has two During the first 22 years, Option B is the better
different retirement investment plans from which to choice because the total is greater than that of
choose. Option A. However, after about 22 years, the
a. Write equations for Option A and Option B given balance of Option A exceeds that of Option B, so
the minimum deposits. Option A is the better choice.
b. Draw a graph to show the balances for each
investment option after t years. 41. MULTIPLE REPRESENTATIONSIn this
problem, you will explore the rapid increase of an
c. Explain whether Option A or Option B is the
better investment choice. 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.
SOLUTION:
c. SYMBOLIC
Use the pattern in the table to write
a. an equation for the number of pieces in the stack
Use the compound interest formula. after x cuts.
The equation that represents Option A d. ANALYTICAL
The thickness of ordinary paper
is . is about 0.003 inch. Write an equation for the
thickness of the stack of paper after x cuts.
e.ANALYTICAL
The equation that represents Option B How thick will the stack of
paper be after 30 cuts?
is
SOLUTION:
b. a.
The graph that shows the balances for each There will be 2, 4, 8, 16 pieces after the first, second,
investment option after t years: third and fourth cut respectively.
b.
c.
The equation that represent the situation is
d.
Substitute 0.003 for a and 2 for b in the exponential
function.
e.
c. Substitute 30 for x in the equation
During the first 22 years, Option B is the better and
choice because the total is greater than that of simplify.
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 The thickness of the stack of paper after 30 cuts is
problem, you will explore the rapid increase of an about 3221225.47 in.
exponential function. A large sheet of paper is cut in
half, and one of the resulting pieces is placed on top
WRITING IN MATHIn a problem about
of the other. Then the pieces in the stack are cut in 42.
half and placed on top of each other. Suppose this compound interest, describe what happens as the
procedure is repeated several times. compounding period becomes more frequent while
a. CONCRETE the principal and overall time remain the same.
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 SOLUTION:
many pieces after the third cut? How many pieces Sample answer: The more frequently interest is
after the fourth cut? compounded, the higher the account balance
b. TABULAR becomes.
Record your results in a table.
x −
c. SYMBOLIC
Use the pattern in the table to write
ERROR ANALYSIS
an equation for the number of pieces in the stack 43. Beth and Liz are solving 6
3 > 36−x − 1. Is either of them correct? Explain your
after x cuts. reasoning.
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?
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.
Substitute 0.003 for a and 2 for b in the exponential
function.
e.
Substitute 30 for x in the equation and
simplify. SOLUTION:
Sample answer: Beth; Liz added the exponents
instead of multiplying them when taking the power of
a power.
18 18 18
CHALLENGESolve for x: 16 + 16 + 16 +
44.
18 18 x
The thickness of the stack of paper after 30 cuts is 16 + 16 = 4 .
about 3221225.47 in.
SOLUTION:
Simplify the exponential equation.
WRITING IN MATHIn a problem about
42.
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. Use a graphing calculator to solve the exponential
x − equation for x.
ERROR ANALYSIS
43. Beth and Liz are solving 6 Enter 4x as Y1 and 2.36118 E22 as Y2. Adjust the
3 > 36−x − 1. Is either of them correct? Explain your viewing window to see both graphs and the
reasoning. intersection.
Use the CALC function to find the intersection of
the two graphs.
So, the solution to the equation is about x =
37.1610.
OPEN ENDEDWhat would be a more beneficial
45.
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%.
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.
−
18 18 18 b. The graph of an exponential growth equation is
CHALLENGESolve for x: 16 + 16 + 16 +
44. increasing.
18 18 x
16 + 16 = 4 .
c. The graph of an exponential decay equation is
increasing.
SOLUTION:
Simplify the exponential equation. SOLUTION:
x 20x
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.
Use a graphing calculator to solve the exponential
OPEN ENDEDWrite an exponential inequality with
equation for x. 47.
a solution of x 2.
Enter 4x as Y1 and 2.36118 E22 as Y2. Adjust the ≤
viewing window to see both graphs and the
SOLUTION:
intersection. x 2
Sample answer: 4 4
Use the CALC function to find the intersection of ≤
the two graphs. 2x x + 1 2x + 2 4x + 1
PROOFShow that 27 · 81 = 3 · 9 .
48.
SOLUTION:
So, the solution to the equation is about x =
37.1610.
WRITING IN MATHIf you were given the initial
49.
OPEN ENDEDWhat would be a more beneficial
45. and final amounts of a radioactive substance and the
change to a 5-year loan at 8% interest compounded amount of time that passes, how would you
monthly: reducing the term to 4 years or reducing the determine the rate at which the amount was
interest rate to 6.5%? increasing or decreasing in order to write an
equation?
SOLUTION:
Reducing the term will be more beneficial. The
multiplier is 1.3756 for the 4-year and 1.3828 for the SOLUTION:
Sample answer: Divide the final amount by the initial
6.5%. amount. If n is the number of time intervals that pass,
take the nth root of the answer.
CCSS ARGUMENTS
46. Determine whether the
following statements are sometimes, always, or −4 =
never true. Explain your reasoning. 50. 3 × 10
A 30,000
a. x 20x −
2 > 8 for all values of x.
− B 0.0003
b. The graph of an exponential growth equation is C 120
increasing. −
D 0.00003
c. The graph of an exponential decay equation is
increasing.
SOLUTION:
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. B is the correct option.
51. Which of the following could not be a solution to 5 −
OPEN ENDEDWrite an exponential inequality with
47. 3x < 3?
a solution of x 2. −
≤
F2.5
SOLUTION:
G3
x 2 H
Sample answer: 4 4 3.5
≤
J
2x x + 1 2x + 2 4x + 1 4
PROOFShow that 27 · 81 = 3 · 9 .
48.
SOLUTION:
SOLUTION: Check the inequality by substituting 2.5 for x.
So, F is the correct option.
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
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,
take the nth root of the answer.
−4 =
50. 3 × 10
A
30,000
− The measure of the smallest angle in the triangle is
B 0.0003 30 .
º
C 120
− SAT/ACT Which of the following is equivalent to
D 0.00003 53.
(x)(x)(x)(x) for all x?
SOLUTION:
A x + 4
B 4x
C 2x2
D 4x2
E 4
x
B is the correct option.
SOLUTION:
51. Which of the following could not be a solution to 5 −
3x < 3?
−
F
2.5
G3 E is the correct choice.
H 3.5 Graph each function.
J
4 x
y = 2(3)
54.
SOLUTION:
Check the inequality by substituting 2.5 for x. SOLUTION:
Make a table of values. Then plot the points and
sketch the graph.
So, F is the correct option.
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
x
y = 5(2)
The measure of the smallest angle in the triangle is 55.
30 . SOLUTION:
º
SAT/ACT Which of the following is equivalent to Make a table of values. Then plot the points and
53. sketch the graph.
(x)(x)(x)(x) for all x?
A x + 4
B 4x
C 2x2
D 4x2
E 4
x
SOLUTION:
E is the correct choice.
Graph each function.
x
y = 2(3)
54.
SOLUTION:
Make a table of values. Then plot the points and
sketch the graph.
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.
Solve each equation.
57.
SOLUTION:
58.
SOLUTION:
56.
59.
SOLUTION:
Make a table of values. Then plot the points and SOLUTION:
sketch the graph.
60.
SOLUTION:
The square root of x cannot be negative, so there is
no solution.
Solve each equation. 61.
SOLUTION:
57.
SOLUTION:
62.
58.
SOLUTION:
SOLUTION:
59.
63.
SOLUTION:
SOLUTION:
60.
SOLUTION:
64.
SOLUTION:
The square root of x cannot be negative, so there is
no solution.
61.
SOLUTION:
65.
SOLUTION:
62.
SOLUTION:
SALESA 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:
Let I be the income of the salesperson and m be his
sales.
The function that represent the situation is
.
63.
Substitute 1000 for I in the equation and solve for m.
SOLUTION:
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
64. third booth received z pounds of each type of cheese.
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?
65. SOLUTION:
The system of equations that represent the situation:
SOLUTION:
Eliminate the variable x by using two pairs of
equations.
SALES A salesperson earns $10 an hour plus a 10% Subtract (1) and (2).
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 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:
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.
By the end of the day, the dairy had sold 131 pounds Substitute z = 100 in the equation
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.
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.
h(x) = x2 + 2
69.
g(x) = x − 3
SOLUTION:
Solve the system of two equations:
h(x) = x2 + 1
70.
g(x) = 2x + 1
Substitute z = 100 in the equation −
SOLUTION:
Substitute y = 150 and z = 100 in the (1) and solve
for x. h(x) = 5x
71. −
g(x) = 3x − 5
SOLUTION:
Booth 1 has 190 lb; Booth 2 has 150 lb; Booth 3 has
100 lb.
h(x) = x3
Find [g h](x) and [h g](x). 72.
◦ ◦
h(x) = 2x 1 g(x) = x − 2
68. −
g(x) = 3x + 4
SOLUTION:
SOLUTION:
73. h(x) = x + 4
g(x) = | x |
SOLUTION:
h(x) = x2 + 2
69.
g(x) = x − 3
SOLUTION:
h(x) = x2 + 1
70.
g(x) = 2x + 1
−
SOLUTION:
h(x) = 5x
71. −
g(x) = 3x − 5
SOLUTION:
h(x) = x3
72.
g(x) = x − 2
SOLUTION:
73. h(x) = x + 4
g(x) = | x |
SOLUTION:
Solve each equation.
5x 2x − 4
1. 3 = 27
SOLUTION:
Use the Property of Equality for Exponential
Functions.
2y − 3 y + 1
2. 16 = 4
SOLUTION:
Use the Property of Equality for Exponential
Functions.
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.
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.
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 .
b.
Substitute 1 for t in the function and solve for c.
x + 5 8x − 6
4. 49 = 7
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.
SCIENCE
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 SOLUTION:
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.
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? 8.
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.
Solve each inequality. Solve each equation.
2x + 6 2x – 4
7. 4 ≤64 4x + 2
9. 8 = 64
SOLUTION:
SOLUTION:
Use the Property of Equality for Exponential
Use the Property of Inequality for Exponential Functions.
Functions.
x − 6
10. 5 = 125
SOLUTION:
8.
SOLUTION:
Use the Property of Equality for Exponential
Functions.
Use the Property of Inequality for Exponential a + 2 3a + 1
Functions. 11. 81 = 3
SOLUTION:
Solve each equation.
4x + 2
9. 8 = 64
Use the Property of Equality for Exponential
SOLUTION: Functions.
Use the Property of Equality for Exponential
Functions. b + 2 2 − 2b
12. 256 = 4
SOLUTION:
x − 6
10. 5 = 125
SOLUTION: Use the Property of Equality for Exponential
Functions.
Use the Property of Equality for Exponential
Functions.
3c + 1 3c − 1
13. 9 = 27
a + 2 3a + 1 SOLUTION:
11. 81 = 3
SOLUTION:
Use the Property of Equality for Exponential
Functions.
Use the Property of Equality for Exponential
Functions.
b + 2 2 − 2b 2y + 4 y + 1
12. 256 = 4 14. 8 = 16
SOLUTION:
SOLUTION:
Use the Property of Equality for Exponential Use the Property of Equality for Exponential
Functions. Functions.
7-2 Solving Exponential Equations and Inequalities
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.
2y + 4 y + 1
14. 8 = 16
SOLUTION:
The exponential function that models the situation
is .
Use the Property of Equality for Exponential
Functions. b.
Substitute 22 for x in the modeled function and solve
for y.
15. CCSS MODELINGIn 2009, My-Lien received Write an exponential function for the graph that
$10,000 from her grandmother. Her parents invested
all of the money, and by 2021, the amount will have passes through the given points.
grown to $16,960. 16. (0, 6.4) and (3, 100)
a. Write an exponential function that could be used to
SOLUTION:
model the money y. Write the function in terms of x, Substitute 100 for y and 6.4 for a and 3 for x into an
the number of years since 2009. exponential function and determine the value of b.
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
exponential function and simplify.
An exponential function that passes through the given
points is .
eSolutions Manual - Powered by Cognero Page3
17. (0, 256) and (4, 81)
SOLUTION:
The exponential function that models the situation Substitute 81 for y and 256 for a and 4 for x into an
is . exponential function and determine the value of b.
b.
Substitute 22 for x in the modeled function and solve
for y.
Write an exponential function for the graph that An exponential function that passes through the given
passes through the given points. points is .
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
points is . An exponential function that passes through the given
points is .
17. (0, 256) and (4, 81) 19. (0, 144), and (4, 21,609)
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 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 An exponential function that passes through the given
points is . .
points is
18. (0, 128) and (5, 371,293) 20. Find the balance of an account after 7 years if $700
SOLUTION: is deposited into an account paying 4.3% interest
Substitute 371293 for y and 128 for a and 5 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 .
19. (0, 144), and (4, 21,609) 21. Determine how much is in a retirement account after
20 years if $5000 was invested at 6.05% interest
SOLUTION: compounded weekly.
Substitute 21609 for y and 144 for a and 4 for x into
an exponential function and determine the value of b. SOLUTION:
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 .
22. A savings account offers 0.7% interest compounded
20. Find the balance of an account after 7 years if $700 bimonthly. If $110 is deposited in this account, what
is deposited into an account paying 4.3% interest will the balance be after 15 years?
compounded monthly.
SOLUTION:
SOLUTION: Use the compound interest formula.
Use the compound interest formula. Substitute $110 for P, 0.007 for r, 6 for n and 15 for t
Substitute $700 for P, 0.043 for r, 12 for n and 7 for t and simplify.
and simplify.
23. A college savings account pays 13.2% annual
21. Determine how much is in a retirement account after interest compounded semiannually. What is the
20 years if $5000 was invested at 6.05% interest balance of an account after 12 years if $21,000 was
compounded weekly. initially deposited?
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 $21,000 for P, 0.132 for r, 2 for n and 12
for t and simplify. for t and simplify.
22. A savings account offers 0.7% interest compounded Solve each inequality.
bimonthly. If $110 is deposited in this account, what
24.
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
and simplify. Use the Property of Inequality for Exponential
Functions.
25.
23. A college savings account pays 13.2% annual SOLUTION:
interest compounded semiannually. What is the
balance of an account after 12 years if $21,000 was
initially deposited?
SOLUTION: Use the Property of Inequality for Exponential
Use the compound interest formula. Functions.
Substitute $21,000 for P, 0.132 for r, 2 for n and 12
for t and simplify.
26.
Solve each inequality. SOLUTION:
24.
SOLUTION:
Use the Property of Inequality for Exponential
Functions. Use the Property of Inequality for Exponential
Functions.
25.
SOLUTION:
27.
Use the Property of Inequality for Exponential SOLUTION:
Functions.
Use the Property of Inequality for Exponential
Functions.
26.
SOLUTION:
28.
SOLUTION:
Use the Property of Inequality for Exponential
Functions.
Use the Property of Inequality for Exponential
Functions.
27.
SOLUTION:
29.
SOLUTION:
Use the Property of Inequality for Exponential
Functions.
Use the Property of Inequality for Exponential
Functions.
28.
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) .
Use the Property of Inequality for Exponential a. Find the temperature of the hot chocolate after 15
Functions. minutes.
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:
29. a.
Substitute 15 for t in the equation and simplify.
SOLUTION:
b.
Substitute 30 for t in the equation and simplify.
c.
Use the Property of Inequality for Exponential Substitute 10 for t in the equation and simplify.
Functions.
So, temperature of the hot chocolate will be below
60°C after 10 minutes.
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
30. SCIENCEA mug of hot chocolate is 90°C at time t in pounds.
= 0. It is surrounded by air at a constant temperature a. If a 45-pound beaver will defend 170 square yards,
of 20°C. If stirred steadily, its temperature in Celsius write an equation for the area a defended by a
−t beaver weighing w pounds.
after t minutes will be y(t) = 20 + 70(1.071) .
a. Find the temperature of the hot chocolate after 15 b. Scientists believe that thousands of years ago, the
minutes. beaver’s ancestors were 11 feet long and weighed
b. Find the temperature of the hot chocolate after 30 430 pounds. Use your equation to determine the area
minutes. defended by these animals.
c.
The optimum drinking temperature is 60°C. Will
the mug of hot chocolate be at or below this SOLUTION:
temperature after 10 minutes? a.
Substitute 170 for y, 45 for b, and 1.31 for x in the
exponential function.
SOLUTION:
a.
Substitute 15 for t in the equation and simplify.
b.
Substitute 30 for t in the equation and simplify. 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.
c.
Substitute 10 for t in the equation and simplify.
So, temperature of the hot chocolate will be below
Solve each equation.
60°C after 10 minutes.
32.
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. SOLUTION:
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. Use the Property of Equality for Exponential
Functions.
SOLUTION:
a.
Substitute 170 for y, 45 for b, and 1.31 for x in the
exponential function.
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
Solve each equation. Functions.
32.
SOLUTION:
34.
SOLUTION:
Use the Property of Equality for Exponential
Functions.
Use the Property of Equality for Exponential
Functions.
33.
SOLUTION:
35.
SOLUTION:
Use the Property of Equality for Exponential
Functions.
Use the Property of Equality for Exponential
Functions.
34.
SOLUTION:
36.
SOLUTION:
Use the Property of Equality for Exponential
Functions.
35.
Use the Property of Equality for Exponential
Functions.
SOLUTION:
37.
Use the Property of Equality for Exponential
Functions.
SOLUTION:
Use the Property of Equality for Exponential
36. Functions.
SOLUTION:
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. a
Use the equation you wrote in part to estimate
the world population in the year 2020. How accurate
do you think the estimate is? Explain your reasoning.
37.
SOLUTION:
SOLUTION: a.
Substitute 4.458 for y, 2.556 for a, and 30 for x in the
exponential function and solve for b.
Use the Property of Equality for Exponential
Functions. The exponential function that model the situation is
.
b.
Substitute 50 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 c. The prediction was about 375 million greater than
billions for 1950 to 1980. Write the equation in terms the actual population.
of x, the number of years since 1950. (Round the d.
value of b to the nearest ten-thousandth.) Substitute 70 for x in the equation and simplify.
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. Because the prediction for 2000 was greater than the
d. a
Use the equation you wrote in part to estimate actual population, this prediction for 2020 is probably
the world population in the year 2020. How accurate even higher than the actual population will be at the
do you think the estimate is? Explain your reasoning. time.
SOLUTION: 39. TREES The diameter of the base of a tree trunk in
a.
Substitute 4.458 for y, 2.556 for a, and 30 for x in the centimeters varies directly with the power of its
exponential function and solve for b. 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
The exponential function that model the situation is 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
b.
base.
Substitute 50 for x in the equation and simplify.
SOLUTION:
a.
The equation that represent the situation is
.
c. The prediction was about 375 million greater than b.
the actual population. Substitute 84 for h in the equation and solve for d.
d.
Substitute 70 for x in the equation and simplify.
The diameter of the General Sherman Tree at its
Because the prediction for 2000 was greater than the base is about 1001 cm.
actual population, this prediction for 2020 is probably
even higher than the actual population will be at the 40. FINANCIAL LITERACYMrs. Jackson has two
time. different retirement investment plans from which to
choose.
39. TREES The diameter of the base of a tree trunk in a. Write equations for Option A and Option B given
centimeters varies directly with the power of its the minimum deposits.
b. Draw a graph to show the balances for each
height in meters. investment option after t years.
a. A young sequoia tree is 6 meters tall, and the c. Explain whether Option A or Option B is the
diameter of its base is 19.1 centimeters. Use this better investment choice.
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:
The equation that represent the situation is a.
Use the compound interest formula.
. The equation that represents Option A
b.
Substitute 84 for h in the equation and solve for d. is .
The equation that represents Option B
is
b.
The diameter of the General Sherman Tree at its The graph that shows the balances for each
base is about 1001 cm. investment option after t years:
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. Explain whether Option A or Option B is the
better investment choice.
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.
SOLUTION:
a.
Use the compound interest formula. 41. MULTIPLE REPRESENTATIONSIn this
The equation that represents Option A problem, you will explore the rapid increase of an
exponential function. A large sheet of paper is cut in
is . half, and one of the resulting pieces is placed on top
of the other. Then the pieces in the stack are cut in
The equation that represents Option B half and placed on top of each other. Suppose this
procedure is repeated several times.
is a. CONCRETE
Perform this activity and count the
b. number of sheets in the stack after the first cut. How
The graph that shows the balances for each many pieces will there be after the second cut? How
investment option after t years: 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
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.
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.
c.
41. MULTIPLE REPRESENTATIONSIn this
problem, you will explore the rapid increase of an The equation that represent the situation is
exponential function. A large sheet of paper is cut in d.
half, and one of the resulting pieces is placed on top Substitute 0.003 for a and 2 for b in the exponential
of the other. Then the pieces in the stack are cut in function.
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 e.
many pieces will there be after the second cut? How Substitute 30 for x in the equation and
many pieces after the third cut? How many pieces simplify.
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. The thickness of the stack of paper after 30 cuts is
d. ANALYTICAL
The thickness of ordinary paper about 3221225.47 in.
is about 0.003 inch. Write an equation for the
thickness of the stack of paper after x cuts.
WRITING IN MATHIn a problem about
42.
e.ANALYTICAL
How thick will the stack of compound interest, describe what happens as the
paper be after 30 cuts? compounding period becomes more frequent while
the principal and overall time remain the same.
SOLUTION:
a. SOLUTION:
There will be 2, 4, 8, 16 pieces after the first, second, Sample answer: The more frequently interest is
third and fourth cut respectively. compounded, the higher the account balance
b. 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.
WRITING IN MATHIn a problem about
42.
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 SOLUTION:
compounded, the higher the account balance Sample answer: Beth; Liz added the exponents
becomes. instead of multiplying them when taking the power of
a power.
x −
ERROR ANALYSIS
43. Beth and Liz are solving 6 18 18 18
3 −x − 1
CHALLENGESolve for x: 16 + 16 + 16 +
> 36 . Is either of them correct? Explain your 44.
18 18 x
reasoning. 16 + 16 = 4 .
SOLUTION:
Simplify the exponential equation.
Use a graphing calculator to solve the exponential
equation for x.
Enter 4x as Y1 and 2.36118 E22 as Y2. Adjust the
viewing window to see both graphs and the
intersection.
Use the CALC function to find the intersection of
the two graphs.
SOLUTION:
Sample answer: Beth; Liz added the exponents
instead of multiplying them when taking the power of So, the solution to the equation is about x =
a power. 37.1610.
18 18 18
CHALLENGESolve for x: 16 + 16 + 16 +
44.
OPEN ENDEDWhat would be a more beneficial
18 18 x 45.
16 + 16 = 4 . change to a 5-year loan at 8% interest compounded
monthly: reducing the term to 4 years or reducing the
SOLUTION: interest rate to 6.5%?
Simplify the exponential equation.
SOLUTION:
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
Use a graphing calculator to solve the exponential never true. Explain your reasoning.
equation for x. a. x 20x
2 > 8 for all values of x.
−
Enter 4x as Y1 and 2.36118 E22 as Y2. Adjust the b. The graph of an exponential growth equation is
viewing window to see both graphs and the increasing.
intersection.
c. The graph of an exponential decay equation is
Use the CALC function to find the intersection of increasing.
the two graphs.
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.
OPEN ENDEDWrite an exponential inequality with
47.
a solution of x 2.
≤
So, the solution to the equation is about x = SOLUTION:
x 2
Sample answer: 4 4
37.1610. ≤
2x x + 1 2x + 2 4x + 1
OPEN ENDED
45. What would be a more beneficial PROOFShow that 27 · 81 = 3 · 9 .
48.
change to a 5-year loan at 8% interest compounded
monthly: reducing the term to 4 years or reducing the SOLUTION:
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%.
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
WRITING IN MATHIf you were given the initial
increasing. 49.
and final amounts of a radioactive substance and the
c. The graph of an exponential decay equation is amount of time that passes, how would you
increasing. determine the rate at which the amount was
increasing or decreasing in order to write an
SOLUTION: equation?
a. x 20x
Always; 2 will always be positive, and 8 will
−
always be negative. SOLUTION:
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,
take the nth root of the answer.
c. Never; by definition the graph will always be
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: D 0.00003
x 2
Sample answer: 4 4
≤
SOLUTION:
2x x + 1 2x + 2 4x + 1
PROOFShow that 27 · 81 = 3 · 9 .
48.
SOLUTION:
B is the correct option.
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
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.
GRIDDED RESPONSEThe three angles of a
52.
−4 = triangle are 3x, x + 10, and 2x − 40. Find the measure
50. 3 × 10 of the smallest angle in the triangle.
A 30,000
−
B 0.0003 SOLUTION:
Sum of the three angles in a triangle is 180 .
C 120 º
−
D 0.00003
SOLUTION:
B is the correct option.
Which of the following could not be a solution to 5
51. − The measure of the smallest angle in the triangle is
3x < 3?
− 30 .
º
F2.5
G3 SAT/ACT Which of the following is equivalent to
H 53.
3.5 (x)(x)(x)(x) for all x?
J
4
A x + 4
SOLUTION: B 4x
Check the inequality by substituting 2.5 for x. C 2x2
2
D 4x
E 4
x
So, F is the correct option. SOLUTION:
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.
E is the correct choice.
SOLUTION: Graph each function.
Sum of the three angles in a triangle is 180 .
º x
y = 2(3)
54.
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
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
SOLUTION:
x
y = 5(2)
55.
SOLUTION:
Make a table of values. Then plot the points and
E is the correct choice. sketch the graph.
Graph each function.
x
y = 2(3)
54.
SOLUTION:
Make a table of values. Then plot the points and
sketch the graph.
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.
Solve each equation.
57.
SOLUTION:
56.
SOLUTION:
Make a table of values. Then plot the points and
sketch the graph.
58.
SOLUTION:
59.
SOLUTION:
Solve each equation.
60.
57.
SOLUTION:
SOLUTION:
The square root of x cannot be negative, so there is
58. no solution.
SOLUTION:
61.
SOLUTION:
59.
SOLUTION:
62.
SOLUTION:
60.
SOLUTION:
63.
SOLUTION:
The square root of x cannot be negative, so there is
no solution.
61.
SOLUTION:
64.
SOLUTION:
62.
SOLUTION:
65.
SOLUTION:
63.
SOLUTION:
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:
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.
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
65. received y pounds of each type of cheese, and the
third booth received z pounds of each type of cheese.
SOLUTION: 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?
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 SOLUTION:
be? The system of equations that represent the situation:
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.
Eliminate the variable x by using two pairs of
equations.
Subtract (1) and (2).
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 Multiply (2) by 3 and (3) by 4 and subtract both the
third booth received z pounds of each type of cheese. 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?
Solve the system of two equations:
SOLUTION:
The system of equations that represent the situation:
Substitute z = 100 in the equation
Eliminate the variable x by using two pairs of
equations.
Substitute y = 150 and z = 100 in the (1) and solve
Subtract (1) and (2). for x.
Multiply (2) by 3 and (3) by 4 and subtract both the
equations.
Booth 1 has 190 lb; Booth 2 has 150 lb; Booth 3 has
100 lb.
Find [g h](x) and [h g](x).
◦ ◦
68. h(x) = 2x − 1
g(x) = 3x + 4
Solve the system of two equations: SOLUTION:
Substitute z = 100 in the equation
2
h(x) = x + 2
69.
g(x) = x − 3
SOLUTION:
Substitute y = 150 and z = 100 in the (1) and solve
for x.
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.
Find [g h](x) and [h g](x).
◦ ◦
68. h(x) = 2x − 1
g(x) = 3x + 4
SOLUTION:
h(x) = 5x
71. −
g(x) = 3x − 5
SOLUTION:
h(x) = x2 + 2
69. h(x) = x3
72.
g(x) = x − 3
g(x) = x − 2
SOLUTION:
SOLUTION:
73. h(x) = x + 4
g(x) = | x |
2
h(x) = x + 1 SOLUTION:
70.
g(x) = 2x + 1
−
SOLUTION:
h(x) = 5x
71. −
g(x) = 3x − 5
SOLUTION:
h(x) = x3
72.
g(x) = x − 2
SOLUTION:
73. h(x) = x + 4
g(x) = | x |
SOLUTION:
Solve each equation.
5x 2x − 4
1. 3 = 27
SOLUTION:
Use the Property of Equality for Exponential
Functions.
2y − 3 y + 1
2. 16 = 4
SOLUTION:
Use the Property of Equality for Exponential
Functions.
6x x − 2
3. 2 = 32
SOLUTION:
Solve each equation.
5x 2x − 4
1. 3 = 27
SOLUTION:
Use the Property of Equality for Exponential
Functions.
Use the Property of Equality for Exponential
Functions.
x + 5 8x − 6
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.
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.
6x x − 2 a. Write an exponential function to represent the
3. 2 = 32 number of cells c after t minutes.
b. If you begin with one Escherichia coli cell, how
SOLUTION: many cells will there be in one hour?
SOLUTION:
a.
The exponential function that represent the number
of cells after t minutes is .
Use the Property of Equality for Exponential b.
Functions. Substitute 1 for t in the function and solve for c.
6. A certificate of deposit (CD) pays 2.25% annual
x + 5 8x − 6 interest compounded biweekly. If you deposit $500
4. 49 = 7 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 Equality for Exponential
Functions.
Solve each inequality.
2x + 6 2x – 4
7. 4 ≤64
SCIENCE
5. Mitosis is a process in which one cell
divides into two. The Escherichia coli is one of the SOLUTION:
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? Use the Property of Inequality for Exponential
Functions.
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.
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.
Solve each equation.
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
Functions.
x − 6
10. 5 = 125
SOLUTION:
8. Use the Property of Equality for Exponential
Functions.
SOLUTION:
a + 2 3a + 1
11. 81 = 3
SOLUTION:
Use the Property of Inequality for Exponential
Functions.
Solve each equation. Use the Property of Equality for Exponential
4x + 2 Functions.
9. 8 = 64
SOLUTION:
b + 2 2 − 2b
12. 256 = 4
Use the Property of Equality for Exponential SOLUTION:
Functions.
x − 6 Use the Property of Equality for Exponential
10. 5 = 125 Functions.
SOLUTION:
Use the Property of Equality for Exponential 3c + 1 3c − 1
Functions. 13. 9 = 27
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.
2y + 4 y + 1
14. 8 = 16
SOLUTION:
b + 2 2 − 2b
12. 256 = 4
SOLUTION:
Use the Property of Equality for Exponential
Functions.
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
3c + 1 3c − 1 grown to $16,960.
13. 9 = 27 a. Write an exponential function that could be used to
SOLUTION: 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 SOLUTION:
Functions. a.
Substitute 16780 for y 10000 for a and 12 for x in the
exponential function and simplify.
2y + 4 y + 1
14. 8 = 16
SOLUTION: The exponential function that models the situation
is .
b.
Substitute 22 for x in the modeled function and solve
Use the Property of Equality for Exponential for y.
Functions.
Write an exponential function for the graph that
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 SOLUTION:
all of the money, and by 2021, the amount will have Substitute 100 for y and 6.4 for a and 3 for x into an
grown to $16,960. exponential function and determine the value of b.
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.
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 .
17. (0, 256) and (4, 81)
SOLUTION:
Substitute 81 for y and 256 for a and 4 for x into an
exponential function and determine the value of b.
The exponential function that models the situation
is .
b.
Substitute 22 for x in the modeled function and solve
for y.
An exponential function that passes through the given
7-2 Solving Exponential Equations and Inequalities
points is .
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:
Substitute 371293 for y and 128 for a and 5 for x into
SOLUTION: an exponential function and determine the value of b.
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 .
An exponential function that passes through the given
points is . 19. (0, 144), and (4, 21,609)
SOLUTION:
17. (0, 256) and (4, 81) Substitute 21609 for y and 144 for a and 4 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 20. Find the balance of an account after 7 years if $700
points is . is deposited into an account paying 4.3% interest
18. (0, 128) and (5, 371,293) compounded monthly.
SOLUTION:
SOLUTION: Use the compound interest formula.
Substitute 371293 for y and 128 for a and 5 for x into Substitute $700 for P, 0.043 for r, 12 for n and 7 for t
an exponential function and determine the value of b. and simplify.
An exponential function that passes through the given 21. Determine how much is in a retirement account after
points is . 20 years if $5000 was invested at 6.05% interest
eSolutions Manual - Powered by Cognero compounded weekly. Page4
19. (0, 144), and (4, 21,609)
SOLUTION:
Use the compound interest formula.
SOLUTION: Substitute $5000 for P, 0.0605 for r, 52 for n and 20
Substitute 21609 for y and 144 for a and 4 for x into for t and simplify.
an exponential function and determine the value of b.
22. A savings account offers 0.7% interest compounded
An exponential function that passes through the given bimonthly. If $110 is deposited in this account, what
points is . will the balance be after 15 years?
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.
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?
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
for t and simplify.
SOLUTION:
Use the compound interest formula.
Substitute $5000 for P, 0.0605 for r, 52 for n and 20
for t and simplify.
Solve each inequality.
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 Property of Inequality for Exponential
Use the compound interest formula. Functions.
Substitute $110 for P, 0.007 for r, 6 for n and 15 for t
and simplify.
25.
SOLUTION:
23. A college savings account pays 13.2% annual
interest compounded semiannually. What is the Use the Property of Inequality for Exponential
balance of an account after 12 years if $21,000 was Functions.
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.
26.
SOLUTION:
Solve each inequality.
24.
SOLUTION:
Use the Property of Inequality for Exponential
Use the Property of Inequality for Exponential Functions.
Functions.
25.
27.
SOLUTION:
SOLUTION:
Use the Property of Inequality for Exponential
Functions.
Use the Property of Inequality for Exponential
Functions.
26.
SOLUTION:
28.
SOLUTION:
Use the Property of Inequality for Exponential
Functions.
Use the Property of Inequality for Exponential
Functions.
27.
29.
SOLUTION:
SOLUTION:
Use the Property of Inequality for Exponential
Functions.
Use the Property of Inequality for Exponential
Functions.
28.
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.
b. Find the temperature of the hot chocolate after 30
Use the Property of Inequality for Exponential minutes.
c.
Functions. 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.
29.
b.
SOLUTION: Substitute 30 for t in the equation and simplify.
c.
Substitute 10 for t in the equation and simplify.
Use the Property of Inequality for Exponential So, temperature of the hot chocolate will be below
Functions. 60°C after 10 minutes.
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
30. SCIENCEA mug of hot chocolate is 90°C at time t beaver weighing w pounds.
= 0. It is surrounded by air at a constant temperature b. Scientists believe that thousands of years ago, the
of 20°C. If stirred steadily, its temperature in Celsius beaver’s ancestors were 11 feet long and weighed
−t 430 pounds. Use your equation to determine the area
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.
b. Find the temperature of the hot chocolate after 30 SOLUTION:
minutes. a.
Substitute 170 for y, 45 for b, and 1.31 for x in the
c.
The optimum drinking temperature is 60°C. Will exponential function.
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.
Solve each equation.
So, temperature of the hot chocolate will be below 32.
60°C after 10 minutes.
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.
33.
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.
Solve each equation.
32.
SOLUTION:
34.
SOLUTION:
Use the Property of Equality for Exponential
Functions.
Use the Property of Equality for Exponential
Functions.
33.
35.
SOLUTION:
SOLUTION:
Use the Property of Equality for Exponential
Functions. Use the Property of Equality for Exponential
Functions.
34.
36.
SOLUTION:
SOLUTION:
Use the Property of Equality for Exponential
Functions.
Use the Property of Equality for Exponential
Functions.
35.
SOLUTION:
37.
SOLUTION:
Use the Property of Equality for Exponential
Functions.
Use the Property of Equality for Exponential
Functions.
36.
SOLUTION:
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
value of b to the nearest ten-thousandth.)
b. Suppose the population continued to grow at that
Use the Property of Equality for Exponential rate. Estimate the population in 2000.
Functions. c. In 2000, the population of the world was about
6.08 billion. Compare your estimate to the actual
population.
d. a
Use the equation you wrote in part to estimate
the world population in the year 2020. How accurate
do you think the estimate is? Explain your reasoning.
SOLUTION:
a.
37. Substitute 4.458 for y, 2.556 for a, and 30 for x in the
exponential function and solve for b.
SOLUTION:
The exponential function that model the situation is
Use the Property of Equality for Exponential .
Functions. b.
Substitute 50 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 c. The prediction was about 375 million greater than
about 4.458 billion. the actual population.
a. x d.
Write an exponential function of the form y = ab Substitute 70 for x in the equation and simplify.
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
rate. Estimate the population in 2000. Because the prediction for 2000 was greater than the
c. In 2000, the population of the world was about actual population, this prediction for 2020 is probably
6.08 billion. Compare your estimate to the actual even higher than the actual population will be at the
population. time.
d. a
Use the equation you wrote in part to estimate 39. TREES The diameter of the base of a tree trunk in
the world population in the year 2020. How accurate
do you think the estimate is? Explain your reasoning. centimeters varies directly with the power of its
SOLUTION: height in meters.
a. a. A young sequoia tree is 6 meters tall, and the
Substitute 4.458 for y, 2.556 for a, and 30 for x in the diameter of its base is 19.1 centimeters. Use this
exponential function and solve for b. 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
The exponential function that model the situation is base.
.
b. SOLUTION:
Substitute 50 for x in the equation and simplify. a.
The equation that represent the situation is
.
b.
Substitute 84 for h in the equation and solve for d.
c. The prediction was about 375 million greater than
the actual population.
d.
Substitute 70 for x in the equation and simplify.
The diameter of the General Sherman Tree at its
base is about 1001 cm.
40. FINANCIAL LITERACYMrs. Jackson has two
Because the prediction for 2000 was greater than the different retirement investment plans from which to
actual population, this prediction for 2020 is probably choose.
even higher than the actual population will be at the a. Write equations for Option A and Option B given
time. the minimum deposits.
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.
centimeters varies directly with the power of its c. Explain whether Option A or Option B is the
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
base.
SOLUTION:
a.
Use the compound interest formula.
SOLUTION:
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.
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. Explain whether Option A or Option B is the
better investment choice.
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
SOLUTION: exponential function. A large sheet of paper is cut in
a. half, and one of the resulting pieces is placed on top
Use the compound interest formula. of the other. Then the pieces in the stack are cut in
The equation that represents Option A half and placed on top of each other. Suppose this
is . procedure is repeated several times.
a. CONCRETE
Perform this activity and count the
The equation that represents Option B number of sheets in the stack after the first cut. How
many pieces will there be after the second cut? How
is many pieces after the third cut? How many pieces
after the fourth cut?
b. b. TABULAR
The graph that shows the balances for each Record your results in a table.
c. SYMBOLIC
investment option after t years: 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
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.
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 c.
Option A is the better choice. The equation that represent the situation is
d.
Substitute 0.003 for a and 2 for b in the exponential
41. MULTIPLE REPRESENTATIONSIn this
problem, you will explore the rapid increase of an function.
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 e.
procedure is repeated several times. Substitute 30 for x in the equation and
a. CONCRETE simplify.
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. The thickness of the stack of paper after 30 cuts is
c. SYMBOLIC
Use the pattern in the table to write about 3221225.47 in.
an equation for the number of pieces in the stack
after x cuts.
WRITING IN MATHIn a problem about
d. ANALYTICAL 42.
The thickness of ordinary paper compound interest, describe what happens as the
is about 0.003 inch. Write an equation for the compounding period becomes more frequent while
thickness of the stack of paper after x cuts. the principal and overall time remain the same.
e.ANALYTICAL
How thick will the stack of
SOLUTION:
paper be after 30 cuts? Sample answer: The more frequently interest is
SOLUTION: compounded, the higher the account balance
a. becomes.
There will be 2, 4, 8, 16 pieces after the first, second,
third and fourth cut respectively. x −
ERROR ANALYSIS
43. Beth and Liz are solving 6
b. 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.
WRITING IN MATHIn a problem about
42.
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
instead of multiplying them when taking the power of
SOLUTION:
Sample answer: The more frequently interest is a power.
compounded, the higher the account balance 18 18 18
CHALLENGESolve for x: 16 + 16 + 16 +
becomes. 44.
18 18 x
x − 16 + 16 = 4 .
ERROR ANALYSIS
43. Beth and Liz are solving 6
3 > 36−x − 1. Is either of them correct? Explain your SOLUTION:
reasoning. Simplify the exponential equation.
Use a graphing calculator to solve the exponential
equation for x.
Enter 4x as Y1 and 2.36118 E22 as Y2. Adjust the
viewing window to see both graphs and the
intersection.
Use the CALC function to find the intersection of
the two graphs.
So, the solution to the equation is about x =
SOLUTION: 37.1610.
Sample answer: Beth; Liz added the exponents
instead of multiplying them when taking the power of
OPEN ENDEDWhat would be a more beneficial
45.
a power. change to a 5-year loan at 8% interest compounded
18 18 18 monthly: reducing the term to 4 years or reducing the
CHALLENGESolve for x: 16 + 16 + 16 +
44. interest rate to 6.5%?
18 18 x
16 + 16 = 4 .
SOLUTION:
SOLUTION: Reducing the term will be more beneficial. The
Simplify the exponential equation. 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
Use a graphing calculator to solve the exponential increasing.
equation for x.
c. The graph of an exponential decay equation is
Enter 4x as Y1 and 2.36118 E22 as Y2. Adjust the increasing.
viewing window to see both graphs and the
intersection. SOLUTION:
a. x 20x
Always; 2 will always be positive, and 8 will
Use the CALC function to find the intersection of −
the two graphs. 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
≤
2x x + 1 2x + 2 4x + 1
So, the solution to the equation is about x = PROOFShow that 27 · 81 = 3 · 9 .
48.
37.1610.
SOLUTION:
OPEN ENDEDWhat would be a more beneficial
45.
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%.
CCSS ARGUMENTS
46. Determine whether the
following statements are sometimes, always, or
WRITING IN MATHIf you were given the initial
never true. Explain your reasoning. 49.
a. x 20x and final amounts of a radioactive substance and the
2 > 8 for all values of x.
− amount of time that passes, how would you
b. The graph of an exponential growth equation is determine the rate at which the amount was
increasing. increasing or decreasing in order to write an
c. The graph of an exponential decay equation is equation?
increasing.
SOLUTION:
SOLUTION: Sample answer: Divide the final amount by the initial
a. x 20x amount. If n is the number of time intervals that pass,
Always; 2 will always be positive, and 8 will
− take the nth root of the answer.
always be negative.
b. Always; by definition the graph will always be −4
increasing even if it is a small increase. 50. 3 × 10 =
A 30,000
−
c. Never; by definition the graph will always be
decreasing even if it is a small decrease. B 0.0003
C 120
−
OPEN ENDEDWrite an exponential inequality with
47. D 0.00003
a solution of x 2.
≤
SOLUTION:
SOLUTION:
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.
51. Which of the following could not be a solution to 5 −
3x < 3?
−
F2.5
G3
H 3.5
J
4
SOLUTION:
Check the inequality by substituting 2.5 for x.
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? So, F is the correct option.
SOLUTION:
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
of the smallest angle in the triangle.
take the nth root of the answer.
SOLUTION:
−4 Sum of the three angles in a triangle is 180 .
= º
50. 3 × 10
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
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 2
4 C 2x
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.
GRIDDED RESPONSEThe three angles of a
52. Graph each function.
triangle are 3x, x + 10, and 2x − 40. Find the measure
x
of the smallest angle in the triangle. y = 2(3)
54.
SOLUTION:
SOLUTION:
Sum of the three angles in a triangle is 180 .
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
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
4 x
E
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.
Graph each function.
x
y = 2(3)
54.
SOLUTION:
Make a table of values. Then plot the points and
sketch the graph.
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.
Solve each equation.
57.
SOLUTION:
56.
58.
SOLUTION:
Make a table of values. Then plot the points and SOLUTION:
sketch the graph.
59.
SOLUTION:
60.
SOLUTION:
Solve each equation.
57.
SOLUTION:
The square root of x cannot be negative, so there is
no solution.
61.
58.
SOLUTION:
SOLUTION:
62.
59.
SOLUTION:
SOLUTION:
60.
63.
SOLUTION:
SOLUTION:
The square root of x cannot be negative, so there is
no solution.
61. 64.
SOLUTION:
SOLUTION:
62.
SOLUTION:
65.
SOLUTION:
63.
SOLUTION: 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:
Let I be the income of the salesperson and m be his
sales.
The function that represent the situation is
.
64. Substitute 1000 for I in the equation and solve for m.
SOLUTION:
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.
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?
SALES A salesperson earns $10 an hour plus a 10% SOLUTION:
66. The system of equations that represent the situation:
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
sales. Eliminate the variable x by using two pairs of
The function that represent the situation is equations.
.
Substitute 1000 for I in the equation and solve for m.
Subtract (1) and (2).
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 Multiply (2) by 3 and (3) by 4 and subtract both the
the beginning of one day, the first booth received x equations.
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 Solve the system of two equations:
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
Eliminate the variable x by using two pairs of for x.
equations.
Subtract (1) and (2).
Booth 1 has 190 lb; Booth 2 has 150 lb; Booth 3 has
100 lb.
Multiply (2) by 3 and (3) by 4 and subtract both the
Find [g h](x) and [h g](x).
equations. ◦ ◦
68. h(x) = 2x − 1
g(x) = 3x + 4
SOLUTION:
Solve the system of two equations:
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. 2
70. h(x) = x + 1
g(x) = 2x + 1
−
SOLUTION:
Booth 1 has 190 lb; Booth 2 has 150 lb; Booth 3 has
100 lb.
Find [g h](x) and [h g](x).
◦ ◦
68. h(x) = 2x − 1
h(x) = 5x
g(x) = 3x + 4 71. −
g(x) = 3x − 5
SOLUTION:
SOLUTION:
h(x) = x3
72.
g(x) = x − 2
2
h(x) = x + 2 SOLUTION:
69.
g(x) = x − 3
SOLUTION:
73. h(x) = x + 4
g(x) = | x |
SOLUTION:
h(x) = x2 + 1
70.
g(x) = 2x + 1
−
SOLUTION:
h(x) = 5x
71. −
g(x) = 3x − 5
SOLUTION:
h(x) = x3
72.
g(x) = x − 2
SOLUTION:
73. h(x) = x + 4
g(x) = | x |
SOLUTION:
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