## Modeling Exponential Growth

#### MODELING EXPONENTIAL GROWTH AND DECAY

 The mathematical model for exponential growth or decay is given by                  f(t) = a-bt   where f(t) = amount or size at time t                 a = initial amount (at t=0)                 b = constant representing the growth or decay factor If b > 1    :  exponential growth If 0

#### Example 1

A town with a population of 6,000 grows 2% per year. Find the population at the end of 10 years.

Solution:

This is exponential growth with a=6,000 (initial population).

The growth factor is b=100% + 2% = 102% = 1.02

Substituting the values of a and b in the exponential growth model f(t) = a-bt : f(t) = 6000(1.02t )

Find f(t) when t=10 years.

f(t)=6000(1.0210) » 7314

At the end of 10 years, the population will be 7314.

#### Example 2

Suppose the acreage of forest is decreasing at 1% each year because of development. If there are currently 4,000,000 acres of forest, determine the amount of forest after 20 years.

Solution:

This is exponential decay with a=4,000,000 (initial acreage).

The decay factor is b=100% - 1% = 99% = 0.99

Substituting the values of a and b in the exponential decay model f(t) = a-bt : f(t) = 4,000,000(0.99t )

Find f(t) when t=20 years.

f(t)=4,000,000(0.9920) » 3,271,628

At the end of 20 years, the forest land will be 3,271,628 acres.

#### Try these problems

QUESTIONS

Use the model for exponential growth and decay to answer each of the questions.

1. A 4-foot tree grows 10% each year. How tall will it be at the end of 5 years?
2. Suppose your parent invested \$2,000 in an account which pays 4% interest compounded annually. Find the account balance after 10 years.
3. A population of 10,000,000 decreases 1.5% annually for 10 years. What is the population at the end of this period?
4. A \$10,000 purchase decreases 8% in value per year. What is the value of the purchase after 5 years?

1. This is exponential growth with a=4 feet (initial height).
The growth factor is b=100% + 10% = 110% = 1.1
Substituting the values of a and b in the exponential growth model  f(t) = a-bt  :  f(t) = 4(1.1t)

Find f(t) when t=5 years.
f(t)=4(1.15) » 6.44 feet
At the end of 5 years, the tree will be 6.44 ft tall.

2. This is exponential growth with a=\$2,000 (initial investment).
The growth factor is b=100% + 4% = 104% = 1.04
Substituting the values of a and b in the exponential growth model  f(t) = a-bt  :  f(t) = 2000(1.04t )

Find f(t) when t=10 years.
f(t)=2000(1.0410) » 2960
At the end of 10 years, the account balance will be \$2,960.

3. This is exponential decay with a=10,000,000 (initial population).
The decay factor is b=100% - 1.5% = 98.5% = 0.985
Substituting the values of a and b in the exponential growth model  f(t) = a-bt  :  f(t) = 10,000,0000.985t  )

Find f(t) when t=10 years.
f(t)=10,000,000(0.98510) » 8,597,304
The population will be approximately 8,597,304 after 10 years.

4. This is exponential decay with a=\$10,000 (initial purchase value).
The decay factor is b=100% - 8% = 92% = 0.92
Substituting the values of a and b in the exponential growth model  f(t) = a-bt  :  f(t) = 10,000(0.92 t )

Find f(t) when t=5 years.
f(t)=10,000(0.925) » \$6591
The value of the purchase after 5 years will be \$6591.