Algebra 2- Exponential Function

Question

The population of Knox is 48,500 and is increasing at a rate of 2.5% per year. Write an expression for the projected population of Knox after n years. Then, predict the population of Knox to the nearest hundred after 8 years.

Answer 1

Hi there -

  So an exponential function is a function that describes a situation where you are multiplying something by the same number over and over again, like the population increase problem you present here.  After 1 year, the population will grow by:

 

48500 * 2.5% =

48500 * 0.025 = 1212.5   (or 1213.. no half people!!)

for a total of 49713 people

 

another way of looking at that is that the total population after one year, will by 102.5% of the population the year before:

48500 * 102.5% =

48500 * 1.025 = 49,712.5 (or 49713, rounded to the nearest whole person)

 

the following year the population will grow by another 2.5%, but this time its 2.5% of 49,712.5, not 48,500.

 

so that's kind of like:

49,712.5 * 1.205

or

48500 * 1.025 * 1.025

 

notice that for 2 years, we have multiplied by 1.025 TWO times.  For 3 years, we'd multiply by 1.025 three times.  For 13 years, 13 times... and so on.

 

We express this mathematically like this:

Starting population * growth rate plus 100% ^{raised to the power of the number of years}

 

For this example:

48500 (1.025ⁿ)

 

 

to find out what the population is after 8 years, we just plug in 8 for 'n'

48500 (1.025⁸)= approx. 59,092.54

 

rounded to the nearest hundred : 59,100