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EXPONENTIAL GROWTH MODEL WRITING EXPONENTIAL GROWTH MODELS A quantity is growing exponentially if it increases by the same percent in each time period. C is the initial amount. t is the time period. (1 + r) is the growth factor, r is the growth rate. The percent of increase is 100r. y = C (1 + r) t

Exponential growth and decay

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Page 1: Exponential growth and decay

EXPONENTIAL GROWTH MODEL

WRITING EXPONENTIAL GROWTH MODELS

A quantity is growing exponentially if it increases by the same percent in each time period.

C is the initial amount. t is the time period.

(1 + r) is the growth factor, r is the growth rate.

The percent of increase is 100r.

y = C (1 + r)t

Page 2: Exponential growth and decay

Finding the Balance in an Account

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

A6 = 500(1.08) 6 793.437

Balance after three years

Balance after six years

Page 3: Exponential growth and decay

EXPONENTIAL GROWTH MODEL

C is the initial amount.

t is the time period.

(1 + r) is the growth factor, r is the growth rate.

The percent of increase is 100r.

y = C (1 + r)t

EXPONENTIAL GROWTH MODEL

500 is the initial amount. 6 is the time period.

(1 + 0.08) is the growth factor, 0.08 is the growth rate.

A6 = 500(1.08) 6 793.437 Balance after 6 years

A6 = 500 (1 + 0.08) 6

SOLUTION METHOD 2 USE A FORMULA

Finding the Balance in an Account

COMPOUND INTEREST You deposit $500 in an account that pays 8% annual interest compounded yearly. What is the account balance after 6 years?

Use the exponential growth model to find the account balance A. The growthrate is 0.08. The initial value is 500.

Page 4: Exponential growth and decay

Writing an Exponential Growth Model

A population of 20 rabbits is released into a wildlife region. The population triples each year for 5 years.

Page 5: Exponential growth and decay

So, the growth rate r is 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%.So, the growth rate r is 2 and the percent of increase each year is 200%.

1 + r = 31 + r = 3

Writing an Exponential Growth Model

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

The population triples each year, so the growth factor is 3.

Reminder: percent increase is 100r.

Page 6: Exponential growth and decay

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?

Writing an Exponential Growth Model

SOLUTION

After 5 years, the population is

P = C(1 + r) t Exponential growth model

= 20(1 + 2) 5

= 20 • 3 5

= 4860

Help

Substitute C, r, and t.

Simplify.

Evaluate.

There will be about 4860 rabbits after 5 years.

Page 7: Exponential growth and decay

A Model with a Large Growth Factor

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.

t

P 486060 180 540 162020

51 2 3 40

0

1000

2000

3000

4000

5000

6000

1 72 3 4 5 6Time (years)

Po

pu

lati

on

P = 20 ( 3 ) t Here, the large

growth factor of 3 corresponds to a rapid increase

Here, the large growth factor of 3 corresponds to a rapid increase

Page 8: Exponential growth and decay

WRITING EXPONENTIAL DECAY MODELS

A quantity is decreasing exponentially if it decreases by the same percent in each time period.

EXPONENTIAL DECAY MODEL

C is the initial amount.t is the time period.

(1 – r ) is the decay factor, r is the decay rate.

The percent of decrease is 100r.

y = C (1 – r)t

Page 9: Exponential growth and decay

Writing an Exponential Decay Model

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?

SOLUTION Let y represent the purchasing power and let t = 0 represent the year 1982. The initial amount is $1. Use an exponential decay model.

= (1)(1 – 0.035) t

= 0.965 t

y = C (1 – r) t

y = 0.96515

Exponential decay model

Substitute 1 for C, 0.035 for r.

Simplify.

Because 1997 is 15 years after 1982, substitute 15 for t.

Substitute 15 for t.

The purchasing power of a dollar in 1997 compared to 1982 was $0.59.

0.59

Page 10: Exponential growth and decay

Graphing the Decay of Purchasing Power

GRAPHING EXPONENTIAL DECAY MODELS

Graph the exponential decay model in the previous example. Use the graph to estimate the value of a dollar in ten years.

SOLUTION Make a table of values, plot the points in a coordinate plane, and draw a smooth curve through the points.

0

0.2

0.4

0.6

0.8

1.0

1 123 5 7 9 11Years From Now

Pu

rch

asin

g P

ow

er

(do

lla

rs)

2 4 6 8 10

t

y 0.8370.965 0.931 0.899 0.8671.00

51 2 3 40

0.70.808 0.779 0.752 0.726

106 7 8 9

Your dollar of today will be worth about 70 cents in ten years.

Your dollar of today will be worth about 70 cents in ten years.

y = 0.965t

Help

Page 11: Exponential growth and decay

GRAPHING EXPONENTIAL DECAY MODELS

EXPONENTIAL GROWTH AND DECAY MODELS

y = C (1 – r)ty = C (1 + r)t

EXPONENTIAL GROWTH MODEL EXPONENTIAL DECAY MODEL

1 + r > 1 0 < 1 – r < 1

CONCEPT

SUMMARY

An exponential model y = a • b t represents exponential

growth if b > 1 and exponential decay if 0 < b < 1.C is the initial amount.t is the time period.

(1 – r) is the decay factor, r is the decay rate.

(1 + r) is the growth factor, r is the growth rate. (0, C)(0, C)

Page 12: Exponential growth and decay

Exponential Growth & Decay Models

• A0 is the amount you start with, t is the time, and k=constant of growth (or decay)

• f k>0, the amount is GROWING (getting larger), as in the money in a savings account that is having interest compounded over time

• If k<0, the amount is SHRINKING (getting smaller), as in the amount of radioactive substance remaining after the substance decays over time

ktoA A t A e

Page 13: Exponential growth and decay

Graphs

00

k

eAA kt

A0A0

0

0

ktA A e

k

Page 14: Exponential growth and decay

Example• Population Growth of the United States. In

1990 the population in the United States was about 249 million and the exponential growth rate was 8% per decade. (Source: U.S. Census Bureau)– Find the exponential growth function.– What will the population be in 2020?– After how long will the population be double

what it was in 1990?

Page 15: Exponential growth and decay

Solution• At t = 0 (1990), the population was about 249 million. We

substitute 249 for A0 and 0.08 for k to obtain the exponential growth function.

A(t) = 249e0.08t

• In 2020, 3 decades later, t = 3. To find the population in 2020 we substitute 3 for t:

A(3) = 249e0.08(3) = 249e0.24 317.

The population will be approximately 317 million in 2020.

Page 16: Exponential growth and decay

Solution continued• We are looking for the doubling time T.

498 = 249e0.08T

2 = e0.08T

ln 2 = ln e0.08T

ln 2 = 0.08T (ln ex = x)

= T

8.7 T

The population of the U.S. will double in about 8.7 decades or 87 years. This will be approximately in 2077.

ln 2

0.08

Page 17: Exponential growth and decay

Exponential Decay• Decay, or decline, is represented by the function A(t) =

A0ekt, k < 0.

In this function:

A0 = initial amount of the substance, A = amount of the substance left after time, t = time, k = decay rate.

• The half-life is the amount of time it takes for half of an amount of substance to decay.

Page 18: Exponential growth and decay

Example Carbon Dating. The radioactive element carbon-14 has a

half-life of 5715 years. If a piece of charcoal that had lost 7.3% of its original amount of carbon, was discovered from an ancient campsite, how could the age of the charcoal be determined?

Solution: The function for carbon dating is

A(t) = A0e-0.00012t.

If the charcoal has lost 7.3% of its carbon-14 from its initial amount A0, then 92.7%A0 is the amount present.

Page 19: Exponential growth and decay

Example continuedTo find the age of the charcoal, we solve the equation for t :

The charcoal was about 632 years old.

0.000120 092.7% tA A e

0.000120.927 te0.00012ln 0.927 ln te

ln 0.927 0.00012t

ln 0.927

0.00012632

t

t