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To find the logistic model, we'll use the formula:

\[ P(t) = \frac{c}{1 + ae^{-bt}} \]

Given dataTo find the logistic model, we'll use the formula:

\[ P(t) = \frac{c}{1 + ae^{-bt}} \]

Given 
- Carrying capacity (\(c\)): 7500
- Initial population (\(P(0)\)): 50
- Population after one week (\(P(7)\)): 200

Let's follow the steps to find \(a\) and \(b\).

1. **Determine \(c\)**:
   The carrying capacity \(c\) is given as 7500.

\[ c = 7500 \]

2. **Find \(a\)** using the initial condition:
   When \(t = 0\), \(P(0) = 50\):

\[ 50 = \frac{7500}{1 + a} \]

Solve for \(a\):

\[ 1 + a = \frac{7500}{50} \]
\[ 1 + a = 150 \]
\[ a = 149 \]

3. **Find \(b\)** using the population after one week:
   When \(t = 7\), \(P(7) = 200\):

\[ 200 = \frac{7500}{1 + 149e^{-7b}} \]

Solve for \(b\):

\[ 1 + 149e^{-7b} = \frac{7500}{200} \]
\[ 1 + 149e^{-7b} = 37.5 \]
\[ 149e^{-7b} = 36.5 \]
\[ e^{-7b} = \frac{36.5}{149} \]
\[ e^{-7b} = \frac{36.5}{149} \]
\[ e^{-7b} = 0.244966 \]
\[ -7b = \ln(0.244966) \]
\[ b = \frac{\ln(0.244966)}{-7} \]

Calculate \(b\):

\[ b \approx \frac{-1.4076}{-7} \]
\[ b \approx 0.2011 \]

So, the parameters for the logistic model are:
\[ c = 7500 \]
\[ a = 149 \]
\[ b = 0.201 \]

Thus, the logistic model is:

\[ P(t) = \frac{7500}{1 + 149e^{-0.201t}} \]

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

1. How do you derive the logistic model formula?
2. What are some real-world applications of logistic models?
3. How does the logistic model differ from exponential growth?
4. What is the significance of the carrying capacity in logistic models?
5. Can you explain the steps to solve for \(a\) and \(b\) in more detail?

**Tip:** Always cross-check your calculations to ensure accuracy, especially when dealing with logarithms and exponential functions.

Learn how to solve the logistic model using initial population and carrying capacity with a step-by-step solution. This problem covers logistic growth concepts, including the logistic model formula and its parameters (c, a, b). Suitable for advanced high school or college students.

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