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 <b <1:  exponential decay

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-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.

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(0.99^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.

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?

ANSWERS

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-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.


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-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.


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-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.


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-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.