Solution:

(A) Find the population function $P(t)$:

The fox population grows at a rate of $8\%$ per year. The general formula for exponential growth is:

$P(t) = P_0 \cdot e^{rt}$

Where:

• $P_0 = 26800$ (initial population in 2000),
• $r = 0.08$ (growth rate as a decimal),
• $t$ = time in years since 2000.

Substitute these values into the formula:

$P(t) = 26800 \cdot e^{0.08t}$

Final Answer for (A):

$P(t) = 26800 \cdot e^{0.08t}$

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(B) Estimate the population in 2008:

The year 2008 corresponds to $t = 8$ (since $t = 0$ is the year 2000). Substitute $t = 8$ into the function:

$P(8) = 26800 \cdot e^{0.08 \cdot 8}$

Simplify the exponent:

$P(8) = 26800 \cdot e^{0.64}$

Using $e^{0.64} \approx 1.8965$:

$P(8) \approx 26800 \cdot 1.8965$

$P(8) \approx 50846$

Final Answer for (B):

The estimated fox population in 2008 is:

$\boxed{50846}$

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Do you want more details on the steps or calculations?

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Relative Questions:

1. What does $e$ represent in exponential growth, and why is it used?
2. How would the function change if the growth rate were 10% instead of 8%?
3. Can this function be used to predict when the population will double? How?
4. What are some limitations of using exponential growth models in ecology?
5. How would the model be adjusted if the population had a carrying capacity?

Tip:

Always double-check your calculations for exponential growth using a calculator or software to ensure precision, especially when approximating constants like $e^{0.64}$.