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To determine the time \( t \) it takes for the population of mice to reach 2000, we need to solve the logistic equation given by:

\[ A(t) = \frac{6000}{1 + 47e^{-0.275t}} \]

We are given \( A(t) = 2000 \). Plugging this into the equation, we get:

\[ 2000 = \frac{6000}{1 + 47e^{-0.275t}} \]

We will solve this equation step by step:

1. **Isolate the exponential term:**
   
\[ 2000 \left( 1 + 47e^{-0.275t} \right) = 6000 \]

2. **Divide both sides by 2000:**

\[ 1 + 47e^{-0.275t} = 3 \]

3. **Subtract 1 from both sides:**

\[ 47e^{-0.275t} = 2 \]

4. **Divide both sides by 47:**

\[ e^{-0.275t} = \frac{2}{47} \]

5. **Take the natural logarithm (ln) of both sides to solve for \( t \):**

\[ -0.275t = \ln \left( \frac{2}{47} \right) \]

6. **Simplify and solve for \( t \):**

\[ t = \frac{\ln \left( \frac{2}{47} \right)}{-0.275} \]

7. **Calculate the value:**

\[ t \approx \frac{\ln (0.0425532)}{-0.275} \]

\[ t \approx \frac{-3.160}{-0.275} \]

\[ t \approx 11.5 \]

Thus, the time it takes for the population to reach 2000 mice is approximately 11.5 months.

Would you like any further details or explanations on any step? Here are some additional questions you might find interesting:

1. How does the logistic equation model population growth?
2. What is the significance of the parameters in the logistic equation?
3. How would the equation change if the carrying capacity were different?
4. Can you explain how to derive the logistic equation?
5. How does the initial population affect the growth curve?

**Tip:** When solving equations involving exponentials and logarithms, always ensure you isolate the exponential term first to simplify the calculation.

Learn how to solve the logistic equation A(t) = 6000 / (1 + 47e^(-0.275t)) to determine the time it takes for a mouse population to reach 2000. This problem involves advanced concepts in exponential functions and logarithms. Suitable for advanced high school or college-level students.

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