Question

2. In 2010, the population of a town is 8500. The population decreases by 4.5% each year.

(a) Write an equation to find the population of the town t years after 2010.
(b) In what year will the population of the town be 7000? Show your work.

Answer 1

This exponential growth/decay (in this case decay because r<1) of the form:

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

Since the population decreases by 4.5% each year the common ratio is:

r=(100-4.5)/100=0.955 so we can say

P(t)=8500(0.955^t)

....

7000=8500(0.955^t)

14/17=(955/1000)^t taking the natural log of both sides

ln(14/17)=t ln(955/1000)

t=ln(14/17)/ln(955/1000)

t≈4.22 years (to nearest hundredth of a year)

Since t is the years since 2010, the population will fall to 7000 in the year (2010+4.22=2014.22, more than four years will have elapsed) 2015.