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To solve the given problem, we need to evaluate the population function \( P(t) \) at \( t = 15 \) and \( t = 16 \) using the formula provided:

\[ P(t) = \frac{600}{1 + 7e^{-0.125t}} \]

### Step-by-Step Solution

#### Step 1: Calculate \( P(15) \)

\[ P(15) = \frac{600}{1 + 7e^{-0.125 \cdot 15}} \]

First, calculate the exponent:

\[ -0.125 \cdot 15 = -1.875 \]

Then, find \( e^{-1.875} \):

\[ e^{-1.875} \approx 0.153 \]

Now, substitute this value back into the formula:

\[ P(15) = \frac{600}{1 + 7 \cdot 0.153} = \frac{600}{1 + 1.071} = \frac{600}{2.071} \approx 289.65 \]

Rounded to the nearest whole number:

\[ P(15) \approx 290 \]

#### Step 2: Calculate \( P(16) \)

\[ P(16) = \frac{600}{1 + 7e^{-0.125 \cdot 16}} \]

First, calculate the exponent:

\[ -0.125 \cdot 16 = -2 \]

Then, find \( e^{-2} \):

\[ e^{-2} \approx 0.135 \]

Now, substitute this value back into the formula:

\[ P(16) = \frac{600}{1 + 7 \cdot 0.135} = \frac{600}{1 + 0.945} = \frac{600}{1.945} \approx 308.48 \]

Rounded to the nearest whole number:

\[ P(16) \approx 308 \]

#### Step 3: Calculate \( P(16) - P(15) \)

\[ P(16) - P(15) = 308 - 290 = 18 \]

### Final Answers

- \( P(15) \approx 290 \)
- \( P(16) - P(15) \approx 18 \)

Do you want more details or have any questions?

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Here are some questions you might want to ask next:
1. How do you derive the exponential decay factor?
2. What is the significance of the constant 600 in the population model?
3. How does the value of \( t \) affect the population \( P(t) \)?
4. Can you explain the steps to find \( e \) raised to any power without a calculator?
5. How does the initial population affect the growth rate in logistic models?

**Tip:** When dealing with exponential functions, remember that the base \( e \) (approximately 2.718) is crucial in natural growth and decay processes.

Explore how to calculate the population growth using the exponential decay model. Detailed step-by-step solution for finding P(15) and P(16) using the formula P(t) = 600 / (1 + 7e^(-0.125t)). Suitable for advanced high school students and anyone interested in exponential functions and numerical computation.

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