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E XPONENTIAL G ROWTH M ODEL

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E XPONENTIAL G ROWTH M ODEL

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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