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The problem asks to find the limit of the population function \( P(t) \) as \( t \to \infty \).

The population model is given as:
\[
P(t) = \frac{6000}{40 + 60e^{-0.03t}}
\]

To find \( \lim_{t \to \infty} P(t) \), we analyze the behavior of the exponential term \( e^{-0.03t} \) as \( t \to \infty \):

- As \( t \to \infty \), \( e^{-0.03t} \to 0 \) because the exponential function decays to 0 for negative exponents.

Substitute \( e^{-0.03t} \to 0 \) into the expression for \( P(t) \):
\[
P(t) = \frac{6000}{40 + 60(0)} = \frac{6000}{40} = 150
\]

Thus, the limit is:
\[
\lim_{t \to \infty} P(t) = 150
\]

The correct answer is **C) 150**.

Would you like more details, or have any questions?

Here are five related questions to expand on this topic:
1. How does the parameter \( e^{-0.03t} \) affect the population growth over time?
2. What is the initial population \( P(0) \)?
3. What happens to the population if the growth rate constant is changed from \( 0.03 \) to a higher value?
4. How would you interpret the asymptotic behavior of the population in real-life scenarios?
5. Can you explain why exponential decay occurs in this model?

**Tip:** Exponential functions are often used in population models because they accurately describe how populations grow or decay over time.

The population on an island is modeled by P(t) = 6000 / (40 + 60e^{-0.03t}) for t ≥ 0, where P(t) is the number of people on the island after t years. What is lim(t → ∞) P(t)?

Learn how to find the limit of the population model P(t) = 6000 / (40 + 60e^{-0.03t}) as t approaches infinity. This problem explores the concept of exponential decay in population models and is suitable for students in Grades 10-12.

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