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2.2 Exponential Decay: Worked Examples

 

Example 1.  Identify each of the following as a growth or decay exponential function.  Identify the growth or decay factor, the growth or decay rate, and the initial value.    

   

a)                             b)  

c)                                d)  

 

SOLUTION.  

 

The general exponential function is of the form , with C as the initial value and a as the growth or decay factor.

 

       a)   The function represents decay since the decay factor  and the decay     

              rate is . The initial value is  

 

       b)   The function represents growth since the growth factor  and the growth      

             rate is . The initial value is  

 

       c)   Since  , the function represents decay with  decay    

             factor  and decay rate . The initial value is  

 

       d)  The function represents decay since the decay factor  and the decay rate is    

            . The initial value is  

 

 

Example 2.  Write a formula for an exponential function with initial value of 2,200 and a decay factor of 0.25.

 

SOLUTION.  

 

 The general exponential function is of the form , with C as the initial value and a as the decay factor, so the function is given by

 

 

 

 

 

 

Example 3.  Write a formula for an exponential function with initial value of 10,000 and halving every time period.

 

SOLUTION.   Since the initial value is halving every time period, the decay rate is 50% and the decay factor is . The function is given by

 

 

 

 

Example 4.  Suppose we were considering the population of a certain community.  Suppose also that 250,000 people lived there in 2000 and that 5% of the population leave every year.   How many people would be living there in 2010?

 

SOLUTION.  

 

The general exponential function is of the form , with C  as the initial value and  as the decay factor. For our example, the decay factor  and the function is given by

 

 

 

For 2010,  and  

 

 

Example 5.  The official estimates of the remaining world oil reserves are 1,000 billion barrels of oil. The estimated rate of depletion is 3% per year. How much oil will be left in 50 years assuming the current rate of depletion?

 

SOLUTION.  

 

Using , with  and decay factor  we have

 

 

 

The official estimate of the daily world oil consumption is 85 million barrels per day.  How many days of oil remain? You do the math!

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