Find P(15), Round to the nearest whole number

To find \( P(15) \) using the given population model \( P(t) = \frac{600}{1 + 7e^{-0.125t}} \), follow these steps:

1. Substitute \( t = 15 \) into the equation:
   \[
   P(15) = \frac{600}{1 + 7e^{-0.125 \times 15}}
   \]

2. Calculate the exponent:
   \[
   -0.125 \times 15 = -1.875
   \]

3. Compute \( e^{-1.875} \):
   \[
   e^{-1.875} \approx 0.153
   \]

4. Substitute this value back into the equation:
   \[
   P(15) = \frac{600}{1 + 7 \times 0.153}
   \]
   \[
   P(15) = \frac{600}{1 + 1.071}
   \]
   \[
   P(15) = \frac{600}{2.071}
   \]

5. Divide to find \( P(15) \):
   \[
   P(15) \approx 289.7
   \]

Rounding to the nearest whole number:
\[
P(15) \approx 290
\]

The population at \( t = 15 \) is approximately 290.

Would you like more details or have any questions? Here are some questions you might consider:

1. How do we compute exponential values like \( e^{-1.875} \)?
2. Why is it important to round the final result?
3. Can this model be used to predict long-term population behavior?
4. What happens to the population as \( t \) increases indefinitely?
5. How does the initial population size affect the model?

**Tip:** When rounding numbers, if the digit after the rounding place is 5 or more, round up. Otherwise, round down.

Learn how to calculate population growth using exponential functions with a step-by-step solution. This problem involves applying the exponential decay formula to find the population at a specific time. Suitable for high school students studying exponential functions.

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