C is the initial amount.

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C is the initial amount.
t is the time period.
y C (1 r)t
(1 r) is the growth factor, r is the growth
rate.
The percent of increase is 100r.
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COMPOUND INTEREST You deposit 500 in an account
that pays 8 annual interest compounded yearly.
What is the account balance after 6 years?
SOLUTION
METHOD 1 SOLVE A SIMPLER PROBLEM
Find the account balance A1 after 1 year and
multiply by the growth factor to find the balance
for each of the following years. The growth rate
is 0.08, so the growth factor is 1 0.08 1.08.
A1 500(1.08) 540
Balance after one year
A2 500(1.08)(1.08) 583.20
Balance after two years
A3 500(1.08)(1.08)(1.08) 629.856
Balance after three years
A6 500(1.08) 6 ?793.437
Balance after six years
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COMPOUND INTEREST You deposit 500 in an account
that pays 8 annual interest compounded yearly.
What is the account balance after 6 years?
SOLUTION
METHOD 2 USE A FORMULA
Use the exponential growth model to find the
account balance A. The growthrate is 0.08. The
initial value is 500.
4
A population of 20 rabbits is released into a
wildlife region. The population triples each year
for 5 years.
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A population of 20 rabbits is released into a
wildlife region. The population triples each
year for 5 years. a. What is the percent of
increase each year?
SOLUTION
The population triples each year, so the growth
factor is 3.
1 r 3
1 r 3
So, the growth rate r is 2 and the percent of
increase each year is 200.
Reminder percent increase is 100r.
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A population of 20 rabbits is released into a
wildlife region. The population triples each
year for 5 years. b. What is the population after
5 years?
Help
SOLUTION
After 5 years, the population is
P C(1 r) t
Exponential growth model
20(1 2) 5
Substitute C, r, and t.
20 3 5
Simplify.
4860
Evaluate.
There will be about 4860 rabbits after 5 years.
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GRAPHING EXPONENTIAL GROWTH MODELS
Graph the growth of the rabbit population.
SOLUTION
Make a table of values, plot the points in a
coordinate plane, and draw a smooth curve through
the points.
Here, the large growth factor of 3 corresponds to
a rapid increase
P 20 ( 3 ) t
8
A quantity is decreasing exponentially if it
decreases by the same percent in each time period.
C is the initial amount.
t is the time period.
y C (1 r)t
(1 r ) is the decay factor, r is the decay rate.
The percent of decrease is 100r.
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COMPOUND INTEREST From 1982 through 1997, the
purchasing powerof a dollar decreased by about
3.5 per year. Using 1982 as the base for
comparison, what was the purchasing power of a
dollar in 1997?
Let y represent the purchasing power and let t
0 represent the year 1982. The initial amount is
1. Use an exponential decay model.
SOLUTION
y C (1 r) t
Exponential decay model
(1)(1 0.035) t
Substitute 1 for C, 0.035 for r.
0.965 t
Simplify.
Because 1997 is 15 years after 1982, substitute
15 for t.
y 0.96515
Substitute 15 for t.
?0.59
The purchasing power of a dollar in 1997 compared
to 1982 was 0.59.
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GRAPHING EXPONENTIAL DECAY MODELS
Help
Graph the exponential decay model in the previous
example. Use the graph to estimate the value of
a dollar in ten years.
Make a table of values, plot the points in a
coordinate plane, and draw a smooth curve through
the points.
SOLUTION
y 0.965t
Your dollar of today will be worth about 70
cents in ten years.
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GRAPHING EXPONENTIAL DECAY MODELS
EXPONENTIAL GROWTH MODEL
EXPONENTIAL DECAY MODEL
y C (1 r)t
y C (1 r)t
An exponential model y a b t represents
exponential growth if b gt 1 and exponential
decay if 0 lt b lt 1.
C is the initial amount.
t is the time period.
0 lt 1 r lt 1
1 r gt 1
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