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The fox population in a certain region has an annual growth rate of 8 percent per year. (Note: Foxes mate once per year.)

It is estimated that the fox population in the year 2020 was 7100.

(a) Find an exponential function that models the fox population \( t \) years after 2020 (Note: \( t = 0 \) for 2020).

The function is \( P(t) = 7100(1 + 0.08)^t \)

(b) Use the function from part (a) to estimate the fox population in the year 2028. (The answer should be an integer.)

There will be 13144 foxes in the year 2028.

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Solution

The problem involves finding an exponential function that models the fox population and using it to estimate the population in a future year. Given: - The annual growth rate is 8%. - The fox population in 2020 (when \( t = 0 \)) was 7100. ### (a) Exponential Function The general formula for exponential growth is: \[ P(t) = P_0 (1 + r)^t \] Where: - \( P_0 = 7100 \) (initial population in 2020), - \( r = 0.08 \) (annual growth rate). Substitute these into the formula: \[ P(t) = 7100 (1 + 0.08)^t \] So the exponential function is: \[ P(t) = 7100 (1.08)^t \] ### (b) Estimate Population in 2028 To find the fox population in 2028, calculate for \( t = 8 \) (since 2028 is 8 years after 2020). \[ P(8) = 7100 (1.08)^8 \] Calculate \( (1.08)^8 \): \[ (1.08)^8 \approx 1.85093 \] Now substitute back to find \( P(8) \): \[ P(8) = 7100 \times 1.85093 \approx 13166.623 \] Rounding to the nearest integer, the estimated fox population in 2028 is 13167.