11-3 Exponential Growth
and Decay
Why learn this?
Exponential growth and decay
describe many real-world
situations, such as the value
of artwork. (See Example 1.)
Objective
Solve problems involving
exponential growth and
decay.
Vocabulary
exponential growth
compound interest
exponential decay
half-life
Exponential growth occurs when a
quantity increases by the same rate
r in each time period t. When this
happens, the value of the quantity
at any given time can be calculated
as a function of the rate and the
original amount.
Exponential Growth
An exponential growth function has the form y = a(1 + r) , where a > 0.
t
y represents the final amount.
a represents the original amount.
r represents the rate of growth expressed as a decimal.
t represents time.
EXAMPLE
1
Exponential Growth
The original value of a painting is $1400, and the value increases by 9%
each year. Write an exponential growth function to model this situation.
Then find the value of the painting in 25 years.
In Example 1, round
to the nearest
hundredth because
the problem deals
with money. This
means you are
rounding to the
nearest cent.
Step 1 Write the exponential growth function for this situation.
y = a (1 + r)
t
Write the formula.
= 1400(1 + 0.09)
= 1400(1.09)
t
t
Substitute 1400 for a and 0.09 for r.
Simplify.
Step 2 Find the value in 25 years.
y = 1400(1.09)t
= 1400(1.09)25
≈ 12,072.31
Substitute 25 for t.
Use a calculator and round to the nearest
hundredth.
The value of the painting in 25 years is $12,072.31.
1. A sculpture is increasing in value at a rate of 8% per year,
and its value in 2000 was $1200. Write an exponential growth
function to model this situation. Then find the sculpture’s
value in 2006.
11-3 Exponential Growth and Decay
805
A common application of exponential growth is compound interest. Recall that
simple interest is earned or paid only on the principal. Compound interest is
interest earned or paid on both the principal and previously earned interest.
Compound Interest
(
r
A=P 1+_
n
)
nt
A represents the balance after t years.
P represents the principal, or original amount.
r represents the annual interest rate expressed as a decimal.
n represents the number of times interest is compounded per year.
t represents time in years.
EXAMPLE
2
Finance Application
Write a compound interest function to model each situation. Then find the
balance after the given number of years.
A $1000 invested at a rate of 3% compounded quarterly; 5 years
Step 1 Write the compound interest function for this situation.
r nt
A=P 1+_
Write the formula.
n
(
)
(
0.03
= 1000 1 + _
4
)
4t
Substitute 1000 for P, 0.03 for r, and 4
for n.
Simplify.
= 1000(1.0075)4t
For compound
interest,
• annually means
“once per year”
(n = 1).
• quarterly means
“4 times per year”
(n = 4).
• monthly means
“12 times per year”
(n = 12).
Step 2 Find the balance after 5 years.
Substitute 5 for t.
A = 1000(1.0075)4(5)
= 1000(1.0075)20
≈ 1161.18
Use a calculator and round to the
nearest hundredth.
The balance after 5 years is $1161.18.
B $18,000 invested at a rate of 4.5% compounded annually; 6 years
Step 1 Write the compound interest function for this situation.
r nt
A=P 1+_
Write the formula.
n
(
(
)
)
0.045
= 18,000 1 + _
1
= 18,000(1.045)
t
t
Substitute 18,000 for P, 0.045 for r, and
1 for n.
Simplify.
Step 2 Find the balance after 6 years.
Substitute 6 for t.
A = 18,000(1.045)6
Use a calculator and round to the
≈ 23,440.68
nearest hundredth.
The balance after 6 years is $23,440.68.
Write a compound interest function to model each situation.
Then find the balance after the given number of years.
2a. $1200 invested at a rate of 3.5% compounded quarterly; 4 years
2b. $4000 invested at a rate of 3% compounded monthly; 8 years
806
Chapter 11 Exponential and Radical Functions
Exponential decay occurs when a quantity decreases by the same rate r in each
time period t. Just like exponential growth, the value of the quantity at any given
time can be calculated by using the rate and the original amount.
Exponential Decay
An exponential decay function has the form y = a (1 - r) , where a > 0.
t
y represents the final amount.
a represents the original amount.
r represents the rate of decay as a decimal.
t represents time.
Notice an important difference between exponential growth functions and
exponential decay functions. For exponential growth, the value inside the
parentheses will be greater than 1 because r is added to 1. For exponential decay,
the value inside the parentheses will be less than 1 because r is subtracted from 1.
EXAMPLE
3
Exponential Decay
The population of a town is decreasing at a rate of 1% per year. In 2000
there were 1300 people. Write an exponential decay function to model this
situation. Then find the population in 2008.
Step 1 Write the exponential decay function for this situation.
In Example 3, round
your answer to
the nearest whole
number because
there can only be
a whole number of
people.
y = a (1 - r)
t
Write the formula.
= 1300(1 - 0.01)
= 1300(0.99)
t
t
Substitute 1300 for a and 0.01 for r.
Simplify.
Step 2 Find the population in 2008.
y = 1300(0.99)8
≈ 1200
Substitute 8 for t.
Use a calculator and round to the nearest
whole number.
The population in 2008 is approximately 1200 people.
3. The fish population in a local stream is decreasing at a rate
of 3% per year. The original population was 48,000. Write an
exponential decay function to model this situation. Then find
the population after 7 years.
A common application of exponential decay is half-life. The half-life of a
substance is the time it takes for one-half of the substance to decay into
another substance.
Half-life
A = P (0.5)
t
A represents the final amount.
P represents the original amount.
t represents the number of half-lives in a given time period.
11-3 Exponential Growth and Decay
807
EXAMPLE
4
Science Application
Fluorine-20 has a half-life of 11 seconds.
A Find the amount of fluorine-20 left from a 40-gram sample
after 44 seconds.
Step 1 Find t, the number of half-lives in the given time period.
44 s = 4
_
11 s
Step 2 A = P(0.5)
Divide the time period by the half-life.
The value of t is 4.
t
Write the formula.
= 40(0.5)
4
Substitute 40 for P and 4 for t.
= 2.5
Use a calculator.
There are 2.5 grams of fluorine-20 remaining after 44 seconds.
B Find the amount of fluorine-20 left from a 40-gram sample after
2.2 minutes. Round your answer to the nearest hundredth.
Step 1 Find t, the number of half-lives in the given time period.
2.2(60) = 132
132 s = 12
_
11 s
Find the number of seconds in 2.2 minutes.
Divide the time period by the half-life.
132
= 12.
The value of t is ___
11
Step 2 A = P (0.5)
Write the formula.
= 40(0.5)12
Substitute 40 for P and 12 for t.
≈ 0.01
Use a calculator. Round to the nearest hundredth.
There is about 0.01 gram of fluorine-20 remaining after 2.2 minutes.
t
4a. Cesium-137 has a half-life of 30 years. Find the amount
of cesium-137 left from a 100-milligram sample after
180 years.
4b. Bismuth-210 has a half-life of 5 days. Find the amount of
bismuth-210 left from a 100-gram sample after 5 weeks.
(Hint: Change 5 weeks to days.)
THINK AND DISCUSS
1. Describe three real-world situations that can be described by
exponential growth or exponential decay functions.
2. The population of a town after t years can be modeled by
t
P = 1000(1.02) . Is the population increasing or decreasing? By what
percentage rate?
3. An exponential function is a function of the form y = ab x. Explain why
both exponential growth functions and exponential decay functions are
exponential functions.
4. GET ORGANIZED Copy and complete the graphic organizer.
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Chapter 11 Exponential and Radical Functions
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