Sure, let's solve this step by step.
### Part (a)
We need to find a function that models the population [tex]\( P(t) \)[/tex] based on the given information.
We know:
- The initial population in 2020 ( [tex]\( t = 0 \)[/tex] ) is 28300.
- The annual growth rate is 6% per year.
The general formula for population growth is:
[tex]\[ P(t) = P_0 \cdot (1 + r)^t \][/tex]
Where:
- [tex]\( P(t) \)[/tex] is the population at time [tex]\( t \)[/tex].
- [tex]\( P_0 \)[/tex] is the initial population.
- [tex]\( r \)[/tex] is the growth rate.
- [tex]\( t \)[/tex] is the number of years after the initial year.
Given [tex]\( P_0 = 28300 \)[/tex] and [tex]\( r = 0.06 \)[/tex], the model becomes:
[tex]\[ P(t) = 28300 \cdot (1.06)^t \][/tex]
Thus, the function that models the fox population [tex]\( t \)[/tex] years after 2020 is:
[tex]\[ P(t) = 28300 \cdot (1.06)^t \][/tex]
### Part (b)
Now, we need to estimate the fox population in the year 2028 using the function from part (a).
Since 2028 is 8 years after 2020, we substitute [tex]\( t = 8 \)[/tex] into our model:
[tex]\[ P(8) = 28300 \cdot (1.06)^8 \][/tex]
Using the function:
[tex]\[ P(8) = 28300 \cdot 1.593848 \approx 28300 \cdot 1.5938 \][/tex]
When we calculate this, we obtain the population in 2028 to be approximately:
[tex]\[ P(8) \approx 45105 \][/tex]
Thus, the estimated fox population in 2028 is:
[tex]\[ \boxed{45105} \][/tex]
