Stewart J. College Algebra: Concepts and Contexts
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254 CHAPTER 3
■
Exponential Functions and Models
Since the initial amount C is 75, the exponential decay model that
we seek is
Model
where x is the number of hours since the drug was administered.
(b) Replacing x by 4 in our model, we get
Replace x by 4
Calculator
So after 4 hours approximately 18 mg of the drug remains.
■ NOW TRY EXERCISE 41 ■
Radioactive elements decay when their atoms spontaneously emit radiation and
become smaller, stable atoms. For instance, uranium-238 emits alpha and beta parti-
cles (two common types of radiation) and decays into nonradioactive lead. Physicists
express the rate of decay in terms of half-life, the time required for half the mass of
the radioactive substance to decay. The half-life of radium-226 is 1600 years, so a
100-g sample decays to 50 g (or g) in 1600 years. This means that the 1600-
year decay factor of radium-226 is .
1
2
1
2
* 100
L 18.008
m14 2= 75
#
10.70 2
4
m 1x2= 75
#
10.702
x
m 1x2= Ca
x
The half-lives of radioactive
elements vary widely.
Element Half-life
Thorium-232 14.5 billion years
Uranium-235 4.5 billion years
Thorium-230 80,000 years
Plutonium-239 24,360 years
Carbon-14 5,730 years
Radium-226 1,600 years
Cesium-137 30 years
Strontium-90 28 years
Polonium-210 140 days
Thorium-234 25 days
Iodine-135 8 days
Radon-222 3.8 days
Lead-211 3.6 minutes
Krypton-91 10 seconds
example
5
An Exponential Decay Model for Radium
The half-life of radium-226 is 1600 years. A 50-gram sample of radium-226 is
placed in an underground disposal facility and monitored.
(a) Find a function that models the mass of radium-226 remaining after x
half-lives.
(b) Use the model to predict the amount of radium-226 remaining after 4000
years.
(c) Make a table of values for with x varying between 0 and 5.
(d) Graph the function and the entries in the table. What does the graph tell
us about how radium-226 decays.
Solution
(a) The initial mass is 50 g, and the decay factor is
1600-year decay factor
So the model that we seek is
where x is the number of 1600-year time periods.
(b) To find the number of time periods that 4000 years represents, we divide:
4000
1600
= 2.5
m 1x2= 50
#
A
1
2
B
x
m1x 2= Ca
x
a =
1
2
m 1x2
m 1x2
m 1x2
figure 2
Graph of m 1x 2= 50
#
A
1
2
B
x
Check your knowledge of the rules of exponents by doing the following prob-
lems. You can review the rules of exponents in
Algebra Toolkits A.3
and
A.4
on
pages T14 and T20.
1. Evaluate each expression.
(a) (b) (c) (d)
2. Simplify each expression.
(a) (b) (c) (d)
3. Write each expression using exponents.
(a) (b) (c) (d)
4. Write each expression using radicals.
(a) (b) (c) (d)
5. Evaluate each expression without using a calculator.
(a) (b) (c) (d)
6. Simplify the expression and eliminate any negative exponent(s). Assume
that all letters denote positive numbers.
(a) (b) (c) (d)
1x
6
y
9
2
-1>3
127x
3
2
2>3
x
x
1>2
x
1>3
x
2>3
A
25
64
B
1>2
8
2>3
121181
4
16
6
2>3
6
1>3
7
-1>2
5
1>2
1
3
5
4
1
3
5
1
13
13
y
6
y
-2
z
5
z
3
15x
3
2
2
x
4
x
3
15
6
2
3
10
8
10
3
4
3
#
4
-5
5
3
#
5
7
SECTION 3.1
■
Exponential Growth and Decay 255
So 4000 years represents 2.5 time periods. Replacing x by 2.5 in the model,
we get
Model
Replace x by 2.5
Calculator
So the mass remaining after 4000 years is about 8.84 g.
(c) We use a calculator to evaluate the entries in the table in the margin.
(d) A graph of the function m and the data points in the table are shown in
Figure 2. From the graph we see that radium-226 decays rapidly at first but
that the rate of decay slows down as time goes on.
■ NOW TRY EXERCISE 45 ■
We see from Example 5 that for an initial amount C of a radioactive substance,
the model
gives the amount remaining after x half-lives.
m 1x2= C A
1
2
B
x
L 8.84
= 50 A
1
2
B
2.5
m 1x 2= 50 A
1
2
B
x
x
m 1x 2
0 50
1 25
2 12.5
3 6.25
4 3.125
5 1.562
m
20
10
40
30
1234 65
0
x
50
7. Which of the following are not models of exponential growth?
(a) (b) (c) (d)
8. Which of the following are not models of exponential decay?
(a) (b) (c) (d)
9–16
■ An exponential growth or decay model is given.
(a) Determine whether the model represents growth or decay.
(b) Find the growth or decay factor.
(c) Find the growth or decay rate.
P 1x 2= 10.5x 2
4
Q 1x 2= 1010.32
x
h 1x 2= 100 - xf 1x 2= 100
#
A
1
5
B
x
Q 1x 2= 410.52
x
h 1t 2= 100xP 1x 2= x
10
f 1x 2= 100
#
7
x
256 CHAPTER 3
■
Exponential Functions and Models
x
f
y
0
x
g
y
0
SKILLS
4. (a) The function models exponential _______ (growth/decay).
(b) The function models exponential
_______ (growth/decay).
Think About It
5. Population A increases by 10% each year, and Population B is multiplied by 1.10 each
year. Compare the growth factors and growth rates of each population. Are they the
same? Explain why or why not.
6. Population A is modeled by and Population B is modeled by
. Explain why both populations decay exponentially. What are the
decay factor and the decay rate for each population?
Q 1x 2= 200
#
3
-x
P 1x 2= 200
#
A
1
3
B
x
,
f 1x 2= 200
#
11.1 2
x
f 1x 2= 50
#
10.25 2
x
CONCEPTS
3.1 Exercises
Fundamentals
1. The number of bacteria in a culture at time t (in hours) is modeled by .
(a) The initial bacteria population is
_______.
(b) The growth factor is
_______.
(c) The growth rate is
_______ (in decimal form).
(d) The growth rate is
_______ (in percentage form).
2. The amount (in grams) of a radioactive substance remaining at time t (years) is modeled
by .
(a) The initial amount is
_______.
(b) The decay factor is
_______.
(c) The decay rate is
_______ (in decimal form).
(d) The decay rate is
_______ (in percentage form).
3. Graphs of exponential functions f and g are shown.
(a) The function f models exponential
_______ (growth/decay).
(b) The function g models exponential
_______ (growth/decay).
f 1t 2= 10010.42
t
N 1t 2= 20011.82
t
SECTION 3.1
■
Exponential Growth and Decay 257
5
12
0
x
10
y
II
10
5
15
1234
0
x
20
y
IV
4
2
8
6
1234
0
x
10
y
I
40
20
60
123
0
x
80
y
III
21–22 ■ A population P is initially 3750 and six hours later reaches the given number. Find
the six-hour growth or decay factor.
21. (a) 7250 (b) 5000 (c) 2500 (d) 750
22. (a) 5800 (b) 9600 (c) 1875 (d) 1250
23–24
■ The graph of a function that models exponential growth or decay is shown, where
x represents one-year time periods. Find the initial population and the one-year
growth factor.
23. 24.
9. 10.
11. 12.
13. 14.
15. 16.
17–20
■ Match the exponential function with its graph.
17. 18.
19. 20. f 1x 2= 511.42
x
f 1x 2= 5
#
A
1
4
B
x
f 1x 2= 10
#
10.5 2
x
f 1x 2= 10
#
2
x
m 1t 2= 3.510.72
t
n 1x 2= 39
#
12.8 2
x
Q 1x 2= 8
#
12.5 2
x
R 1x 2= 1110.252
x
h 1T 2= 10.15 2
T
y = A
1
3
B
x
g1t 2= 20
#
11.9 2
t
f 1x 2= 50
#
11.2 2
x
100
(1, 200)
1
0
x
f
y
y
300
(1, 100)
1
0
x
g
258 CHAPTER 3
■
Exponential Functions and Models
25–34
■ A population P is initially 350. Find an exponential growth model in terms of the
number of time periods x if in each time period the population P
25. quadruples. 26. doubles.
27. decreases by . 28. decreases by .
29. decreases by 35%. 30. increases by 100%.
31. increases by 300%. 32. decreases by 2%.
33. is multiplied by 1.7. 34. is multiplied by 2.3.
35. Bacterial Infection The bacteria Streptococcus pneumoniae (S. pneumoniae) is the
cause of many human diseases, the most common being pneumonia. A culture of these
bacteria initially has 10 bacteria and increases at a rate of 26% per hour.
(a) Find the hourly growth factor a, and find an exponential model for the
bacteria count in the sample, where t is measured in hours.
(b) Use the model you found to predict the number of bacteria in the sample after
5 hours.
36. Bacterial Infection The bacteria R. sphaeroides is the cause of the disease
chlamydia. A culture of these bacteria initially has 25 bacteria and increases at a rate of
28% per hour.
(a) Find the hourly growth factor a, and find an exponential model for the
bacteria count in the sample, where t is measured in hours.
(b) Use the model you found to predict the number of bacteria in the sample after
7 hours.
37. Cost-of-Living Increase Mariel teaches science at a community college in Arizona.
Her starting salary in 2003 was $38,900, and each year her salary increases by 2%.
(a) Complete the table of Mariel’s salary.
f 1t 2= Ca
t
f 1t 2= Ca
t
1
3
1
2
Yea r
Years
since 2003
Salary ($)
2003
0 38,900
2004
2005
2006
2007
(b) Find an exponential growth model for Mariel’s salary t years after 2003. What is
the growth factor?
(c) Use the model found in part (b) to predict Mariel’s salary in 2010, assuming that
her salary continues to increase at the same rate.
(d) Use a graphing calculator to graph the model you found in part (b) together with a
scatter plot of the data in the table.
38. Investment Value Kerri invested $2000 on January 1, 2005, in a mutual fund that for
the past 5 years had a yearly increase of 5%. Assume that the mutual fund continues to
grow at 5% each year.
(a) Complete the following table for the investment value at the beginning of each year.
CONTEXTS
Laurence Gough/Shutterstock.com 2009
SECTION 3.1
■
Exponential Growth and Decay 259
Year
Years
since 2005
Investment value ($)
2005
0 2000
2006
2007
2008
(b) Find an exponential growth model for Kerri’s investment t years since 2005. What
is the growth factor?
(c) Use the model found in part (b) to predict Kerri’s investment value in 2010.
(d) Use a graphing calculator to graph the model you found in part (b) together with a
scatter plot of the data in the table.
39. Health-Care Spending The Centers for Medicare and Medicaid Services report that
health-care expenditures per capita were $2813 in 1990 and $3329 in 1993. Assume that
this rate of growth continues.
(a) Find the three-year growth factor a, and find an exponential growth model
for the annual health-care expenditures per capita, where x is the
number of three-year time periods since 1990.
(b) Use the model found in part (a) to predict health-care expenditures per capita in
1996 and in 2005.
(c) Search the Internet to find the actual health-care expenditures for 1996 and 2005.
Were your predictions reasonable?
40. Profit Increase In 2007, Mieko’s software company made a profit of $5,165,000.
Mieko projects that her company will have a profit of 6 million dollars in 2011. Assume
that her predictions are correct and that this rate of growth continues.
(a) Find the four-year growth factor a, and find an exponential growth model
for the annual profit, where x is the number of four-year time periods
since 2007.
(b) Use the model found in part (a) to predict the profit for Mieko’s company in 2015.
41. Drug Absorption A patient is administered 100 mg of a therapeutic drug. It is known
that 25% of the drug is expelled from the body each hour.
(a) Find an exponential decay model for the amount of drug remaining in the patient’s
body after t hours.
(b) Use the model to predict the amount of the drug that remains in the patient’s body
after 6 hours.
42. Drug Absorption A patient is administered 250 mg of a therapeutic drug. It is known
that 40% of the drug is expelled from the body each hour.
(a) Find an exponential decay model for the amount of drug remaining in the patient’s
body after t hours.
(b) Use the model to predict the amount of the drug that remains in the patient’s body
after 10 hours.
43. Declining Housing Prices In 2006 the U.S. “housing bubble” was beginning to
burst. One news article at the time stated: “The median price of a home sold in the
United States in the third quarter of 2006 was $232,300, and forecasters predict that the
median price of houses will fall by 8.9% each year.” Assume that the median price
decreases at the forecasted decay rate of .- 8.9%
P 1x 2= Ca
x
E 1x 2= Ca
x
R
R
260 CHAPTER 3
■
Exponential Functions and Models
(a) Find the decay factor a, and find an exponential growth model for the
median price of a home sold in the United States x years since the prediction.
(b) Use the model found in part (a) to predict the median price of homes sold in the
third quarter of 2009.
44. Falling Gas Prices A California newspaper published a story on October 17,
2008, stating: “The average price of a gallon of gas has dropped to $3.60, which is a
drop of 10% in just one week. Forecasters predict that this trend will continue and
the price of gas will be under $2.00 a gallon by the end of November.” Assume that
the price of gas decreases at the forecasted decay rate of per week.
(a) Find the decay factor a, and find an exponential growth model for the
price of gas x weeks since October 17, 2008, when the price was $3.60 per gallon.
(b) Use the model to predict the price of gas 6 weeks later. Is this what the
forecasters predicted?
45. Radioactive Cesium The half-life of cesium-137 is 30 years. Suppose we have a
10-g sample.
(a) Find a function that models the mass of cesium-137 remaining after x
half-lives.
(b) Use the model to predict the amount of cesium-137 remaining after 270 years.
(c) Make a table of values for with x varying between 0 and 5.
(d) Graph the function and the entries in the table. What does the graph tell us
about how cesium-137 decays?
46. Radioactive Iodine The half-life of iodine-135 is 8 days. Suppose we have a
22-mg sample.
(a) Find a function that models the mass of iodine-135 remaining after x half-
lives.
(b) Use the model to predict the amount of iodine-135 remaining after 32 days.
(c) Make a table of values for with x varying between 0 and 5.
(d) Graph the function and the entries in the table. What does the graph tell us
about how iodine-135 decays?
47. Grey Squirrel Population The American grey squirrel is a species not native to
Great Britain that was introduced there in the early twentieth century and has been
increasing in numbers ever since. The graph shows the grey squirrel population in a
British county between 1990 and 2000. Assume that the population continues to
grow exponentially.
(a) What was the grey squirrel population in 1990?
(b) Find the ten-year growth factor a, and find an exponential growth model
, where x is the number of ten-year time periods since 1990.
(c) Use the model found in part (b) to predict the grey squirrel population in 2010.
P 1x 2= Ca
x
m 1x 2
m 1x 2
m 1x 2
m 1x 2
m 1x 2
m 1x 2
G 1x 2= Ca
x
- 10%
E 1x 2= Ca
x
P
100,000
(10, 285,000)
123456789101112
Years since 1
99
0
Grey squirrel
population
0
x
SECTION 3.2
■
Exponential Models: Comparing Rates 261
48. Red Squirrel Population The red squirrel is the native squirrel of Great Britain, but
its numbers have been declining ever since the American grey squirrel was introduced
to Great Britain. The graph shows the red squirrel population in a British county
between 1990 and 2000. Assume that the population continues to decrease
exponentially.
(a) What was the red squirrel population in 1990?
(b) Find the ten-year decay factor a, and find an exponential decay model
for the red squirrel population, where x is the number of ten-year time periods since
1990.
(c) Use the model found in part (b) to predict the red squirrel population in 2010.
P 1x 2= Ca
x
P
20,000
(10, 13,000)
2 4 6 8 10 12 14 16 18 20
Years since 1990
0
x
Red squirrel
population
49. Algebra and Alcohol After alcohol is fully absorbed into the body, it is metabolized
with a half-life of 1.5 hours. Suppose Thad consumes 45 mL of alcohol (ethanol).
(a) Find an exponential decay model for the amount of alcohol remaining after x
1.5-hour time intervals.
(b) Graph the function you found in part (a) for x between 0 and 4.
2
3.2 Exponential Models: Comparing Rates
■
Changing the Time Period
■
Growth of an Investment: Compound Interest
IN THIS SECTION… we find exponential models of growth and decay for different time
periods. We also study compound interest as an example of exponential growth.
GET READY… by reviewing how to solve power equations in Algebra Toolkit C.1. Test
your understanding by doing the Algebra Checkpoint on page 267.
To determine the growth rate of a certain type of bacteria, a biologist prepares a
nutrient solution in which the bacteria can grow. The biologist introduces a num-
ber of bacteria into the solution and then estimates the bacteria count 40 minutes
later. From this information the biologist can determine the 40-minute growth fac-
tor. But the biologist wants to report the hourly growth factor, not the 40-minute
growth factor. This conforms to the standards for reporting bacteria growth rates
262 CHAPTER 3
■
Exponential Functions and Models
2
■ Changing the Time Period
A researcher may measure the growth of an insect population daily but may want to
know the weekly growth factor. To do this, suppose that the daily growth factor is a.
In a week the population undergoes seven daily time periods, so the weekly growth
factor is . Similarly, if the weekly growth factor is b, then the daily growth factor
a must satisfy , so .a = b
1>7
a
7
= b
a
7
Growth factor
Daily: a
Weekly: a
7
Weekly: b
Daily: b
1>7
Monthly: c
Yearly: c
12
Yearly: d
Monthly: d
1>12
example
1
Changing the Time Period
A biologist finds that the 30-minute growth rate for a certain type of bacteria is 0.85.
Find the one-hour growth rate.
Solution
The 30-minute growth factor is . Let a be the one-hour growth fac-
tor. Since there are two 30-minute time periods in one hour, we have
30-minute growth factor is 1.85
Calculator
So the one-hour growth factor is 3.4. It follows that the one-hour growth rate is
.
■ NOW TRY EXERCISE 9 ■
r = a - 1 = 3.4 - 1 = 2.4
a L 3.4
a = 1.85
2
1 + 0.85 = 1.85
example
2
Changing the Time Period
A chinchilla farm starts with 20 chinchillas, and after 3 years there are 128 chin-
chillas. Assume that the number of chinchillas grows exponentially.
(a) Find the 3-year growth factor. (b) Find the one-year growth factor.
Solution
(a) The 3-year growth factor is 6.4 as shown in Example 2 of Section 3.1 (page 251).
(b) To find the one-year growth factor, we first let a be the one-year growth factor.
Then after 3 years the population is multiplied by a three times. So the 3-year
growth factor is . Since we know that the 3-year growth factor is 6.4, we can
find a as follows:
a
3
and allows scientists to directly compare the growth rates for different types of
bacteria. So the situation is as follows:
■
We know the 40-minute growth factor.
■
We want to find the one-hour growth factor.
In this section we study how to change growth factors and growth rates from one
time period to a different time period.
SECTION 3.2
■
Exponential Models: Comparing Rates 263
In Example 3 we solve the
power equation .
Solving such equations is
reviewed in Algebra Toolkit C.1,
page T47.
a
3
ⴝ 6.4
example
3
Models for Different Time Periods
A bacterial infection starts with 100 bacteria, and the bacteria count doubles every
five-hour time period. Find an exponential growth model for the number of bacteria
(a) x time periods after infection. (b) t hours after infection.
Solution
(a) The initial number of bacteria is 100, and the growth factor for a five-hour
time period is 2. So a model for the number of bacteria is
where x is the number of five-hour time periods since infection.
(b) We first let a be the one-hour growth factor. Then is the five-hour growth
factor. From part (a) we know that the five-hour growth factor is 2. So
The five-hour growth factor is 2
Solve for a
Calculator
So the one-hour growth factor a is 1.15. So an exponential model is obtained as
follows:
Model
Replace C by 100, a by 1.15
The model is , where t is measured in hours.
Another solution Let t be the number of hours since infection. Since each time
periods consists of 5 hours, after x time periods we have t = 5x. We solve this
equation for x:
Each time period x is 5 h
Divide by 5 and switch sides
To obtain an exponential model in terms of one-hour time periods, we replace
x by t>5 in the model in part (a):
Replace x by t>5
Property of exponents
Calculator
where t is measured in hours.
■ NOW TRY EXERCISE 13 ■
L 100
#
11.15 2
t
= 100
#
12
1>5
2
t
P 1t 2= 100
#
2
t>5
x =
t
5
t = 5x
P 1t2= 10011.15 2
t
P 1t2= 10011.15 2
t
P 1t2= Ca
t
L 1.15
a = 2
1>5
a
5
= 2
a
5
P 1x2= 100
#
2
x
3-year growth factor is 6.4
Take cube root
Calculator
So the one-year growth factor is 1.86.
■ NOW TRY EXERCISES 5 AND 45 ■
a L 1.86
a = 16.42
1>3
a
3
= 6.4