Answer :
iven:
heere are given that the population in the year 2000 was 12000 and the growth rate is 7% per year.
xplanation:
ccording to the question:
For t =0 which is the year 2000:
[tex]P(0)=12000[/tex]For t = 1:
[tex]P(1)=12000+7\%(12000)[/tex]If we say r is the rate, then:
[tex]\begin{gathered} P(1)=12000+r(12000) \\ P(1)=12000(1+r) \end{gathered}[/tex]Then,
For t = 2:
[tex]\begin{gathered} P(2)=12000(1+r)(1+r) \\ P(2)=12000(1+r)^2 \end{gathered}[/tex]And,
For t = 3:
[tex]P(3)=12000(1+r)^3[/tex]Therefore our function should be:
(a):
he population function:
[tex]P(t)=12000(1.07)^t[/tex]Now,
(b):
ccording to the question:
2008 is 8 year from year 2000:
Therefore, t = 8:
Then,
Put the value 8 for t into the function (a):
[tex]\begin{gathered} P(8)=12000(1.07)^t \\ P(8)=12000(1.07)^8 \\ P(8)=12000(1.718) \\ P(8)=20616 \end{gathered}[/tex]inal answer:
[tex]P(t)=12000\times(1.07)^{t-2000}[/tex]