Question

A city population, which was initially 35,000, has been dropping by 3.5% a year. Write an exponential function and graph the function. Use the graph to predict when the population will drop below 15,000.

Answer 1

This is an exponential function and can be expressed as:

f=ir^t, f=final amount, i=initial amount, r=common ratio or "rate", t=time common ratio is applied or "time".

In this case: i=35000, r=(100-3.5)/100=0.965, so the exponential decay function is:

f=35000(0.965)^t

You can graph this with any graphing calculator or even if you type in most browser bars "y=35000(0.965^x)"

But mathematically:

15000>35000(0.965)^t  divide both sides by 35000

3/7>0.965^t  take the natural log of both sides

ln(3/7)>t ln(0.965)  dividing both sides by ln(0.965), and remembering to reverse inequality sign because of division/multiplication by a negative value

t>ln(3/7)/ln(0.965)

t>23.78

So the population will drop below 15000 during the 23rd year.