4.2 Constant Percentage Change
We continue to work with exponential functions having formula f ( x ) = Pa x where P is the
initial amount of output and a is the growth or decay factor.
Growth Factor: a = 1 + r
Where r is the constant percentage change.
Decay Factor: a = 1 – r
Example 1
d
The exponential function N = 3500 (1.77) , where d is measured in decades, gives the
number of individuals in a certain population.
(a) What is the constant percent growth rate represented by this function.
(b) Calculate N (1.5) and explain what your answer means.
(c) What is the yearly percentage growth rate?
(d) When will the population reach 15000?
Example 2
At age 25 you start to work for a company and are offered two rather fanciful retirement
options. Option 1: When you retire, you will be paid a lump sum of $25000 for each year of
service. Option 2: When you start to work, the company will deposit $10000 into an account
that pays 1% per month. When you retire, the account will be closed and the balance given
to you.
(a) Calculate the amount you would get at retirement at age 65 for both options.
(b) What is the youngest age of retirement at which option 2 is the better plan?
Example 3
A tank of water is contaminated with 60 pounds of salt. In order to bring the salt
concentration down to a level consistent with EPA standards, clean water is being pumped
into the tank, and the well-mixed overflow is being collected for removal to a toxic waste site.
The result is that at the end of each hour there is 22% less salt in the tank than at the
beginning of the hour. Let S = S(x) denote the number of pounds of salt in the tank x hours
after the flushing begins.
(a) Give a formula for S.
(b) Make a graph of S that show the process for the first 15 hours.
(c) Suppose the cleanup procedure costs $8000 per hour to operate. How much does it
cost to reduce the amount of salt to 3 pounds?
Example 4
When light strikes the surface of a medium such as glass or water, its intensity decreases with
depth. The Beer-Lambert-Bouguer Law states that the percentage of decrease is the same for
each additional unit of depth. In a certain lake, intensity decreases about 75% for each
additional meter of depth.
(a) Explain why intensity I is an exponential function of depth d in meters.
(b) Use a formula to express intensity I as a function of the depth in this lake. Use I0 to
represent the initial intensity.
(c) At what depth will the intensity be one-tenth of the intensity of the light striking the
surface of the lake?
Example 5
The half-life of a certain radioactive substance is 14.5 hours.
(a) Find the hourly decay factor for this substance.
(b) What is the constant percentage change for this substance?