Download 29 Exponential Growth and Decay

29 Exponential Growth and Decay
Warm Up
Simplify each expression.
1. (4 + 0.05)2
2. 25(1 + 0.02)3
3.
4. The first term of a geometric sequence is 3
and the common ratio is 2. What is the 5th
term of the sequence?
5. The function f(x) = 2(4)x models an insect
population after x days. What is the
population after 3 days?
Holt McDougal Algebra 1
29 Exponential Growth and Decay
Objective
Solve problems involving exponential
growth and decay.
Exponential growth occurs when an quantity
increases by the same rate r in each 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.
Holt McDougal Algebra 1
29 Exponential Growth and Decay
Example 1: Exponential Growth
The original value of a painting is $9,000
and the value increases by 7% each year.
Write an exponential growth function to
model this situation. Then find the
painting’s value in 15 years.
Step 1 Write the exponential growth function
for this situation.
Write the formula.
Substitute 9000 for a and
0.07 for r.
Simplify.
Step 2 Find the value in 15 years.
Substitute 15 for t.
Use a calculator and round
to the nearest
hundredth.
The value of the painting in 15 years is
Holt McDougal Algebra 1
29 Exponential Growth and Decay
Example 2
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.
Step 1 Write the exponential growth function
for this situation.
Write the formula
Substitute 1200 for a and
0.08 for r.
Simplify.
Step 2 Find the value in 6 years.
Substitute 6 for t.
Use a calculator and round
to the nearest
hundredth.
The value of the painting in 6 years is
Holt McDougal Algebra 1
29 Exponential Growth and Decay
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.
Reading Math
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).
Holt McDougal Algebra 1
29 Exponential Growth and Decay
Example 3: Finance Application
Write a compound interest function to model
each situation. Then find the balance after
the given number of years.
$1200 invested at a rate of 2% compounded
quarterly; 3 years.
Step 1 Write the compound interest function for
this situation.
Write the formula.
Substitute 1200 for P, 0.02
for r, and 4 for n.
Simplify.
Holt McDougal Algebra 1
29 Exponential Growth and Decay
Example 3 Continued
Write a compound interest function to model
each situation. Then find the balance after
the given number of years.
$1200 invested at a rate of 2% compounded
quarterly; 3 years.
Step 2 Find the balance after 3 years.
Substitute 3 for t.
Use a calculator and round
to the nearest hundredth.
The balance after 3 years is
Holt McDougal Algebra 1
29 Exponential Growth and Decay
Example 4: Finance Application
Write a compound interest function to model
each situation. Then find the balance after
the given number of years.
$15,000 invested at a rate of 4.8%
compounded monthly; 2 years.
Step 1 Write the compound interest function for
this situation.
Write the formula.
Substitute 15,000 for P,
0.048 for r, and 12 for n.
Simplify.
Holt McDougal Algebra 1
29 Exponential Growth and Decay
Example 4 Continued
Write a compound interest function to model
each situation. Then find the balance after
the given number of years.
$15,000 invested at a rate of 4.8%
compounded monthly; 2 years.
Step 2 Find the balance after 2 years.
Substitute 2 for t.
Use a calculator and round
to the nearest hundredth.
The balance after 2 years
Holt McDougal Algebra 1
29 Exponential Growth and Decay
Example 5
Write a compound interest function to model
each situation. Then find the balance after the
given number of years.
$1200 invested at a rate of 3.5%
compounded quarterly; 4 years
Step 1 Write the compound interest function for
this situation.
Write the formula.
Substitute 1,200 for P, 0.035
for r, and 4 for n.
Simplify.
Holt McDougal Algebra 1
29 Exponential Growth and Decay
Example 5 Continued
Write a compound interest function to model
each situation. Then find the balance after the
given number of years.
$1200 invested at a rate of 3.5%
compounded quarterly; 4 years
Step 2 Find the balance after 4 years.
Substitute 4 for t.
Use a calculator and round
to the nearest hundredth.
The balance after 4 years is
Holt McDougal Algebra 1
29 Exponential Growth and Decay
Example 6
Write a compound interest function to model
each situation. Then find the balance after the
given number of years.
$4000 invested at a rate of 3%
compounded monthly; 8 years
Step 1 Write the compound interest function for
this situation.
Write the formula.
Substitute 4,000 for P, 0.03
for r, and 12 for n.
Simplify.
Holt McDougal Algebra 1
29 Exponential Growth and Decay
Example 6 Continued
Write a compound interest function to model
each situation. Then find the balance after the
given number of years.
$4000 invested at a rate of 3%
compounded monthly; 8 years
Step 2 Find the balance after 8 years.
Substitute 8 for t.
Use a calculator and round
to the nearest hundredth.
The balance after 4 years is
Holt McDougal Algebra 1