A population of deer inside a park has a carrying capacity of 200 and a growth rate of 2. If the initial population is 50 deers, what is the population of deer at any given time? Make a proper and complete solution and make sure its correct and direct to the point no explanation write direct into a paper.

The problem can be solved using the **logistic growth model**. The logistic growth model is given by the equation:

\[
P(t) = \frac{K}{1 + \left( \frac{K - P_0}{P_0} \right) e^{-rt}}
\]

Where:
- \( P(t) \) is the population at time \( t \),
- \( K \) is the carrying capacity (200 deer),
- \( P_0 \) is the initial population (50 deer),
- \( r \) is the growth rate (2).

### Step-by-step Solution:

1. **Substitute the given values** into the equation:
\[
P(t) = \frac{200}{1 + \left( \frac{200 - 50}{50} \right) e^{-2t}}
\]

2. Simplify the expression:
\[
P(t) = \frac{200}{1 + 3 e^{-2t}}
\]

This is the population \( P(t) \) at any given time \( t \).

Let me know if you'd like further details or have any questions.

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Here are 5 related questions to expand on this problem:
1. What happens to the population as \( t \to \infty \)?
2. How would a different growth rate affect the solution?
3. Can you calculate \( P(t) \) for a specific \( t \) (e.g., \( t = 5 \))?
4. How does the logistic growth model differ from exponential growth?
5. What is the significance of the carrying capacity in this model?

**Tip:** Logistic growth models are useful when population growth is limited by resources.

A population of deer inside a park has a carrying capacity of 200 and a growth rate of 2. If the initial population is 50 deer, what is the population of deer at any given time? Make a proper and complete solution and make sure it's correct and direct to the point no explanation write direct into a paper.

Understand how to apply the logistic growth model to calculate the population of deer in a park with a carrying capacity of 200 and an initial population of 50. The step-by-step solution covers key concepts in population dynamics and is suitable for Grades 11-12.

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