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To solve the given logistic growth model for the population of shrews, we need to identify the initial population and the carrying capacity from the given equation:

\[ P(t) = \frac{9600}{1 + 74e^{-0.2t}} \]

### Initial Population:
The initial population is \( P(0) \). To find \( P(0) \), we substitute \( t = 0 \) into the equation:

\[ P(0) = \frac{9600}{1 + 74e^{-0.2 \cdot 0}} \]
\[ P(0) = \frac{9600}{1 + 74 \cdot e^0} \]
\[ P(0) = \frac{9600}{1 + 74 \cdot 1} \]
\[ P(0) = \frac{9600}{75} \]
\[ P(0) = 128 \]

So, the initial population is 128 shrews.

### Carrying Capacity:
The carrying capacity is the maximum population size that the environment can sustain indefinitely. In a logistic model, this is the numerator of the fraction, which in this case is 9600.

So, the carrying capacity is 9600 shrews.

### Summary:
- The initial population is 128 shrews.
- The carrying capacity is 9600 shrews.

Would you like more details or have any questions?

Here are 5 questions you can ask next:
1. How do you derive the formula for logistic growth?
2. Can you explain the significance of each parameter in the logistic growth equation?
3. How does the population change over time according to the logistic growth model?
4. What are some real-world examples of populations that follow logistic growth?
5. How can we determine the rate of growth at a specific time?

**Tip:** In a logistic growth model, the population grows rapidly at first, then slows down as it approaches the carrying capacity. This is due to limited resources becoming a constraint on further growth.

Learn how to solve the logistic growth model for the population of shrews. Understand the initial population and carrying capacity with a step-by-step solution. Suitable for advanced high school students studying population dynamics.

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