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The question in the image asks for the correct model to represent the population of deer growing at a rate of 8% per year, starting from an initial population of 500 in 2016.

To model population growth with a constant percentage increase, we use the exponential growth formula:

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

Where:
- \(P(t)\) is the population at time \(t\),
- \(P_0\) is the initial population (500 in this case),
- \(r\) is the growth rate (8% = 0.08),
- \(t\) is the number of years after the starting point (2016).

Thus, the equation becomes:

\[
P(t) = 500 (1 + 0.08)^t = 500 (1.08)^t
\]

The correct model is:

\[
P(t) = 500 (1.08)^t
\]

From the given choices, the third option ( \( P(t) = 500 (1.08)^t \) ) is the correct one.

Would you like more details or have any questions? Here are five related questions to expand on this concept:

1. How would the model change if the growth rate were 5% instead of 8%?
2. What is the population of deer after 5 years using the given model?
3. How does the exponential growth formula differ from linear growth?
4. Can exponential decay be modeled similarly? How would the formula change for a decay rate?
5. What happens to the population if the growth rate is negative?

**Tip:** Always make sure to express percentage growth as a decimal in the exponential growth formula!

In 2016, a population of deer in an area was estimated to be 500 with a growth rate of 8% each year. Which of the following models estimates the population after t years?

Understand how to model the exponential growth of a deer population starting from 500 with an 8% annual growth rate. Learn the application of exponential growth formulas suitable for middle to high school students.

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