The given information is:
- The anjual growth rate of the fox population is 9% per year.
- The population in the year 2000 was 14500.
a. Find the function that models the population t years after 2000 (t=0 for 2000).
The population growth can be modeled by using the following formula:
[tex]P(t)=A(1+r)^t[/tex]
Where P(t) is the population after t years, A is the initial population and r is the population growth rate in decimal form.
Therefore, we know A=14500, r=9%/100%=0.09.
By replacing these values in the formula, we obtain:
[tex]\begin{gathered} P(t)=14500(1+0.09)^t \\ P(t)=14500(1.09)^t \end{gathered}[/tex]
b. Estimate the fox population in the year 2008.
First, we need to find t for 2008. As t=0 for 2000, then:
[tex]\begin{gathered} t=2008-2000 \\ t=8 \end{gathered}[/tex]
Now, replace t=8 in the formula and find P(8):
[tex]\begin{gathered} P(8)=14500(1.09)^8 \\ P(8)=14500*1.99 \\ P(8)=28892.2 \\ P(8)\approx28893 \end{gathered}[/tex]
The fox population in 2008 is 28893
