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The problem gives the function \( f(x) = 500(1.10)^x \), which models the growth of a fish population in a lake, where \( x \) is the time in months since January 2019. The question asks what the number 500 represents.

In the function, the general form \( f(x) = P(1 + r)^x \) is an exponential growth equation, where:
- \( P \) is the **initial population**,
- \( r \) is the **growth rate**,
- \( x \) is the time (in months in this case).

Here, the number 500 represents the **initial population of fish in the lake**. This is because \( 500 \) is the value of \( P \) in the general form, which indicates the starting number of fish at time \( x = 0 \), which corresponds to January 2019.

So, the correct answer is:
- **The starting number of fish in the lake**.

Would you like further explanation? Here are 5 questions to expand on this:

1. How does the 1.10 factor in the function affect the growth of the fish population?
2. What would the population be after 6 months according to the given model?
3. How can you determine the monthly growth rate from the function?
4. What does the exponent \( x \) represent in terms of time?
5. How does exponential growth differ from linear growth in population modeling?

**Tip:** When interpreting an exponential growth model, always pay close attention to the base of the exponent for growth rate and the coefficient for the initial population.

The growth of a species of fish in a lake can be modeled by the function f(x) = 500(1.10)^x, where x is time in months since January 2019. What does the 500 represent?

Learn how to interpret the exponential growth function f(x) = 500(1.10)^x, where 500 represents the initial fish population. This problem focuses on exponential growth models and is suitable for students in Grades 8-10.

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