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.

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