where f(t) = amount or size at time t If b > 1 : exponential growth |

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-b^{t} : f(t) = 6000(1.02^{t} )

Find f(t) when t=10 years.

f(t)=6000(1.0210) » 7314

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

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(0.99^{t} )

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.

**QUESTIONS**

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

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

**ANSWERS**

- 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-b^{t}: f(t) = 4(1.1^{t})

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. - 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-b^{t}: f(t) = 2000(1.04^{t})

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. - 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-b^{t}: f(t) = 10,000,0000.985^{t})

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. - 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-b^{t}: 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.

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