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The problem asks for a mathematical model to represent the fox population in a region with an annual growth rate of 4%, starting from 21,000 in the year 2000. Let's go through both parts of the problem:

### Part (a) — Finding the Function

The growth rate of the fox population is exponential, as it's increasing by a fixed percentage each year.

The general form for an exponential growth model is:
\[
P(t) = P_0 \times (1 + r)^t
\]
Where:
- \( P(t) \) is the population after \( t \) years,
- \( P_0 \) is the initial population (21,000 in this case),
- \( r \) is the growth rate (4% per year, or 0.04),
- \( t \) is the number of years after the year 2000.

So, the correct function should be:
\[
P(t) = 21000 \times (1 + 0.04)^t = 21000 \times (1.04)^t
\]

### Part (b) — Estimating Population in 2008

To estimate the fox population in the year 2008, we plug in \( t = 8 \) (since 2008 is 8 years after 2000) into the function from part (a):

\[
P(8) = 21000 \times (1.04)^8
\]

Let me calculate that for you:

\[
P(8) = 21000 \times (1.04)^8 \approx 21000 \times 1.3686 \approx 28,740
\]

So, the estimated fox population in 2008 is approximately **28,740**.

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Do you want more details or have any questions?

Here are some follow-up questions based on this information:
1. What would the population be in the year 2015?
2. How long will it take for the population to double?
3. If the growth rate were 3% instead of 4%, how would the function change?
4. What happens if the growth rate becomes negative, say, a decline of 2% per year?
5. How can you graph this function to show the population trend over time?

**Tip:** Exponential growth is a powerful model that applies to many real-world situations, including population dynamics, finance, and more. Understanding its structure helps predict future trends.

The fox population in a certain region has an annual growth rate of 4 percent per year. It is estimated that the population in the year 2000 was 21,000. (a) Find a function that models the population t years after 2000 (t = 0 for 2000). (b) Use the function from part (a) to estimate the fox population in the year 2008.

Learn how to model the exponential growth of a fox population with a 4% annual growth rate starting from 21,000 in the year 2000. This example explains how to find the population in 2008 using the formula P(t) = P0 * (1 + r)^t, where r is 0.04.

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