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.

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.