W RITING E XPONENTIAL G ROWTH M ODELS. A quantity is growing exponentially if it increases by the same percent in each time period. E XPONENTIAL G ROWTH M ODEL. C is the initial amount. t is the time period. y = C (1 + r ) t.
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WRITING EXPONENTIAL GROWTH MODELS
A quantity is growing exponentially if it increases by the same percent in each time period.
EXPONENTIAL GROWTH MODEL
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.
Finding the Balance in an Account
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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 1SOLVE 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)6793.437
Balance after six years
Finding the Balance in an Account
EXPONENTIAL GROWTH MODEL
EXPONENTIAL GROWTH MODEL
C is the initial amount.
t is the time period.
500 is the initial amount.
6 is the time period.
y = C (1 + r)t
A6 = 500 (1 + 0.08)6
(1 + r) is the growth factor,r is the growth rate.
(1 + 0.08) is the growth factor,0.08 is the growth rate.
The percent of increase is 100r.
A6 = 500(1.08)6793.437Balance after6years
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 2USE A FORMULA
Use the exponential growth model to find the account balance A. The growthrate is 0.08. The initial value is 500.
Writing an Exponential Growth Model
A population of 20 rabbits is released into a wildlife region. The population triples each year for 5 years.
Writing an Exponential Growth Model
The population triples each year, so thegrowth factoris 3.
1 + r = 3
So, the growth rate ris 2 and the percent of increase each year is 200%.
So, the growth rate r is 2 and the percent of increase each year is 200%.
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 thegrowth factoris 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.
Writing an Exponential Growth Model
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
SubstituteC,r, andt.
= 20 • 35
Simplify.
= 4860
Evaluate.
There will be about 4860 rabbits after 5 years.
A Model with a Large Growth Factor
t
0
1
2
3
4
5
6000
P
20
60
180
540
1620
4860
5000
4000
Population
3000
2000
1000
0
1
2
3
4
5
6
7
Time (years)
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
WRITING EXPONENTIAL DECAY MODELS
EXPONENTIAL DECAY MODEL
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.
Writing an Exponential Decay Model
COMPOUND INTERESTFrom 1982 through 1997, the purchasing powerof a dollar decreased by about3.5% 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 lett = 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
Substitute1forC,0.035forr.
= 0.965t
Simplify.
Because 1997 is 15 years after 1982, substitute 15 for t.
y = 0.96515
Substitute 15fort.
0.59
The purchasing power of a dollar in 1997 compared to 1982 was $0.59.
Graphing the Decay of Purchasing Power
t
0
1
2
3
4
5
6
7
8
9
10
y
1.00
0.965
0.931
0.899
0.867
0.837
0.808
0.779
0.752
0.726
0.7
1.0
0.8
0.6
Purchasing Power (dollars)
0.4
0.2
0
1
2
3
4
5
6
7
8
9
10
11
12
Years From Now
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.
CONCEPT
EXPONENTIAL GROWTH AND DECAY MODELS
SUMMARY
(1 + r) is the growth factor,r is the growth rate.
(1 – r) is the decay factor,r is the decay rate.
(0,C)
(0,C)
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•btrepresents exponential growth ifb > 1 and exponential decay if 0 < b < 1.
C is the initial amount.
t is the time period.
0 < 1 – r < 1
1 + r > 1