Let's break this problem into two parts as per the question.
### Part (a) - Finding the model function for population growth
Given:
- The initial population in the year 2000, [tex]\( P_0 = 10100 \)[/tex]
- Continuous growth rate, [tex]\( r = 0.07 \)[/tex] (7 percent per year)
We need to find a function [tex]\( P(t) \)[/tex] that models the population [tex]\( t \)[/tex] years after 2000.
The formula for exponential growth is:
[tex]\[ P(t) = P_0 \cdot e^{rt} \][/tex]
Plugging in the given values:
[tex]\[ P(t) = 10100 \cdot e^{0.07t} \][/tex]
So, the function that models the population [tex]\( t \)[/tex] years after 2000 is:
[tex]\[ P(t) = 10100 \cdot e^{0.07t} \][/tex]
### Part (b) - Estimating the fox population in the year 2008
To find the population in the year 2008, we need to evaluate the function [tex]\( P(t) \)[/tex] at [tex]\( t = 8 \)[/tex] (since 2008 is 8 years after 2000).
[tex]\[ P(8) = 10100 \cdot e^{0.07 \cdot 8} \][/tex]
Using the provided numerical result:
[tex]\[ P(8) \approx 17681.79225299062 \][/tex]
As the result needs to be an integer:
The estimated fox population in the year 2008 is approximately:
[tex]\[ \boxed{17682} \][/tex]
