Certainly! Let's work through the given question step-by-step in detail.
### Part (a): Finding the Population Function
We are given that the fox population in the year 2020 is 5800 and it grows at a rate of 7% per year. We need to find a function that models the fox population [tex]\( P(t) \)[/tex], where [tex]\( t \)[/tex] is the number of years after 2020.
Since it’s an exponential growth model, we use the exponential growth formula:
[tex]\[ P(t) = P_0 (1 + r)^t \][/tex]
Here:
- [tex]\( P_0 \)[/tex] is the initial population, which is 5800.
- [tex]\( r \)[/tex] is the growth rate, which is 7% or 0.07.
- [tex]\( t \)[/tex] is the time in years after 2020.
Therefore, the function that models the fox population [tex]\( P(t) \)[/tex] is:
[tex]\[ P(t) = 5800 \times (1 + 0.07)^t \][/tex]
or
[tex]\[ P(t) = 5800 \times 1.07^t \][/tex]
So the answer to part (a) is:
[tex]\[ P(t) = 5800 \times 1.07^t \][/tex]
### Part (b): Estimating the Population in 2028
To estimate the population in the year 2028, we need to determine [tex]\( t \)[/tex] for 2028. Since [tex]\( t = 0 \)[/tex] corresponds to the year 2020:
[tex]\[ t = 2028 - 2020 = 8 \][/tex]
Now, using the function [tex]\( P(t) = 5800 \times 1.07^t \)[/tex], we substitute [tex]\( t = 8 \)[/tex]:
[tex]\[ P(8) = 5800 \times 1.07^8 \][/tex]
Given the computed results, we know this evaluates to approximately:
[tex]\[ P(8) \approx 9965.479843025141 \][/tex]
Rounding to the nearest integer, the estimated fox population in the year 2028 is:
[tex]\[ \boxed{9965} \][/tex]
So the answer to part (b) is:
There will be [tex]\(\boxed{9965}\)[/tex] foxes in the year 2028.
