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The image shows a problem about modeling population growth with an exponential function. Here's a breakdown of the solution:

### Problem:
- The fox population in a region grows at an annual rate of 8%.
- In the year 2000, the population was 16,400.
- We are asked to:
  1. Find a function that models the population \( P(t) \), where \( t \) is the number of years after 2000.
  2. Estimate the fox population in 2008 using this model.

### (a) Finding the Population Function:
The population grows at a constant rate of 8% per year, which can be modeled using the exponential growth formula:

\[
P(t) = P_0 \times (1 + r)^t
\]

where:
- \( P_0 = 16400 \) (initial population in 2000),
- \( r = 0.08 \) (growth rate of 8%),
- \( t \) is the number of years since 2000.

Thus, the function becomes:

\[
P(t) = 16400 \times (1.08)^t
\]

### (b) Estimating the Population in 2008:
In 2008, \( t = 8 \) years since 2000. Plugging \( t = 8 \) into the function:

\[
P(8) = 16400 \times (1.08)^8
\]

Calculating this:

\[
P(8) = 16400 \times 1.85093 \approx 30356.42
\]

Since the problem asks for an integer value, we round to the nearest whole number:

\[
P(8) \approx 30356
\]

This is why the answer shown is around 30,357, but it should be rounded down to 30,356.

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Would you like further details or have any questions?

Here are five related questions to explore:
1. How would the model change if the growth rate were 10% instead of 8%?
2. What would the population be in 2020, using the same model?
3. How long would it take for the population to double?
4. How would you modify the function if there was a yearly decline instead of growth?
5. Can this model be used indefinitely, or are there limitations to this type of growth in real-world scenarios?

**Tip:** Exponential growth models are useful for short-term predictions but may need adjustments for long-term forecasts, as factors like environmental limits can affect growth rates.

The fox population in a certain region has an annual growth rate of 8 percent per year. It is estimated that the population in the year 2000 was 16,400. (a) Find a function that models the population P(t) years after 2000, where t = 0 for 2000. (b) Use the function from part (a) to estimate the fox population in the year 2008. The answer should be an integer.

Learn how to model the growth of a fox population with an 8% annual growth rate, starting at 16,400 in the year 2000. Includes a step-by-step solution to estimate the population in 2008 using exponential growth.

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