Let me guide you through the problem step-by-step:
### Part (a)
We need to find a function that models the population \( P(t) \), where \( t \) is the number of years after 2000.
Given:
- The initial population in the year 2000 (which we consider as \( t = 0 \)) is \( 19400 \).
- The annual growth rate is 5%.
Since the fox population grows exponentially, we can use the formula for exponential growth:
[tex]\[ P(t) = P_0 \times (1 + r)^t \][/tex]
where:
- \( P(t) \) is the population after \( t \) years.
- \( P_0 \) is the initial population.
- \( r \) is the growth rate.
- \( t \) is the time in years after the initial time.
Substituting the given values:
- \( P_0 = 19400 \)
- \( r = 0.05 \)
The function that models the population \( t \) years after 2000 is:
[tex]\[ P(t) = 19400 \times (1 + 0.05)^t \][/tex]
### Part (b)
Now, we need to estimate the fox population in the year 2008.
First, calculate \( t \) for the year 2008:
[tex]\[ t = 2008 - 2000 = 8 \][/tex]
Next, use the function found in part (a):
[tex]\[ P(t) = 19400 \times (1 + 0.05)^t \][/tex]
Substitute \( t = 8 \) into the equation:
[tex]\[ P(8) = 19400 \times (1 + 0.05)^8 \][/tex]
After evaluating the function, we get the population in the year 2008:
[tex]\[ P(8) \approx 28663 \][/tex]
So, the estimated fox population in the year 2008 is approximately:
[tex]\[ \boxed{28663} \][/tex]
