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To solve the problems related to the logistic growth model provided, let's analyze the equation and calculate the required values step by step.

Given the logistic growth model:
\[ P(t) = \frac{2400}{1 + 24e^{-0.128t}} \]
where \( P(t) \) represents the population at time \( t \) in years.

### 1. Number of Caribou Initially Planted
To find the initial number of caribou, we need to evaluate \( P(t) \) at \( t = 0 \):
\[ P(0) = \frac{2400}{1 + 24e^{-0.128 \cdot 0}} = \frac{2400}{1 + 24e^0} = \frac{2400}{1 + 24} = \frac{2400}{25} = 96 \]
So, the initial number of caribou planted was **96 caribou**.

### 2. Carrying Capacity for Caribou on the Island
The carrying capacity is the maximum population that the environment can sustain, which is the limit of \( P(t) \) as \( t \) approaches infinity:
\[ \lim_{t \to \infty} P(t) = \frac{2400}{1 + 24e^{-0.128t}} \]
As \( t \to \infty \), \( e^{-0.128t} \to 0 \):
\[ \lim_{t \to \infty} P(t) = \frac{2400}{1 + 0} = 2400 \]
So, the carrying capacity for the caribou on the island is **2400 caribou**.

### 3. Population After 8 Years
To find the population after 8 years, we substitute \( t = 8 \) into the logistic growth model:
\[ P(8) = \frac{2400}{1 + 24e^{-0.128 \cdot 8}} = \frac{2400}{1 + 24e^{-1.024}} \]
First, we calculate \( e^{-1.024} \):
\[ e^{-1.024} \approx 0.3597 \]
Then,
\[ P(8) = \frac{2400}{1 + 24 \cdot 0.3597} = \frac{2400}{1 + 8.6328} = \frac{2400}{9.6328} \approx 249 \]
So, the population after 8 years is approximately **249 caribou**.

### 4. Time When Population Reaches 1000
To find the time when the population reaches 1000, we set \( P(t) = 1000 \) and solve for \( t \):
\[ 1000 = \frac{2400}{1 + 24e^{-0.128t}} \]
Rearranging and solving for \( e^{-0.128t} \):
\[ 1000(1 + 24e^{-0.128t}) = 2400 \]
\[ 1 + 24e^{-0.128t} = \frac{2400}{1000} \]
\[ 1 + 24e^{-0.128t} = 2.4 \]
\[ 24e^{-0.128t} = 2.4 - 1 \]
\[ 24e^{-0.128t} = 1.4 \]
\[ e^{-0.128t} = \frac{1.4}{24} \]
\[ e^{-0.128t} = 0.0583 \]
Taking the natural logarithm on both sides:
\[ -0.128t = \ln(0.0583) \]
\[ -0.128t \approx -2.841 \]
\[ t \approx \frac{2.841}{0.128} \approx 22.2 \]
So, the population reaches 1000 after approximately **22 years**.

### Summary
1. **Initial number of caribou planted**: 96 caribou
2. **Carrying capacity**: 2400 caribou
3. **Population after 8 years**: 249 caribou
4. **Time to reach 1000 caribou**: 22 years

Would you like further details or have any questions?

Here are five more questions you might be interested in:
1. How do you derive the logistic growth equation?
2. Can you explain the significance of the carrying capacity in logistic models?
3. How does the rate constant affect the growth of the population in the logistic model?
4. What are the applications of logistic growth models in real-life scenarios?
5. How can the initial population size influence the overall growth pattern?

**Tip:** In logistic growth models, the population growth rate slows down as it approaches the carrying capacity due to limited resources. This is a key characteristic differentiating logistic growth from exponential growth.

Understand and solve logistic growth model problems step-by-step with this detailed guide. Explore topics including initial population, carrying capacity, population after a specific time, and time to reach a certain population level. Suitable for advanced high school or college-level students.

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