In an effort to control vegetation overgrowth, 115115 rabbits are released in an isolated area free of predators.  After 11 year, it is estimated that the rabbit population has increased to 345345.  Assuming exponential population growth, what will the population be after another 66 months?  Round to the nearest rabbit.

To solve this problem, we can use the exponential growth model formula:

\[
P(t) = P_0 \times e^{kt}
\]

Where:
- \( P(t) \) is the population at time \( t \).
- \( P_0 \) is the initial population.
- \( k \) is the growth rate constant.
- \( t \) is the time in years.
- \( e \) is the base of the natural logarithm.

### Step 1: Find the Growth Rate Constant \( k \)

We know the following:
- Initial population (\( P_0 \)): 115115 rabbits
- Population after 11 years (\( P(11) \)): 345345 rabbits

Plugging these into the exponential growth formula gives:

\[
345345 = 115115 \times e^{11k}
\]

To solve for \( k \), we first divide both sides by 115115:

\[
\frac{345345}{115115} = e^{11k}
\]

\[
3 = e^{11k}
\]

Now, take the natural logarithm of both sides:

\[
\ln(3) = 11k
\]

\[
k = \frac{\ln(3)}{11}
\]

### Step 2: Calculate the Growth Rate \( k \)

\[
k \approx \frac{1.0986}{11} \approx 0.099873
\]

### Step 3: Find the Population After Another 66 Months

66 months is equivalent to 5.5 years.

Total time from the start is:

\[
t = 11 + 5.5 = 16.5 \text{ years}
\]

Now, use the exponential growth formula again with \( t = 16.5 \) years:

\[
P(16.5) = 115115 \times e^{0.099873 \times 16.5}
\]

### Step 4: Calculate the Final Population

\[
P(16.5) = 115115 \times e^{1.6489055}
\]

\[
P(16.5) = 115115 \times 5.2036 \approx 599209.67
\]

Rounding to the nearest rabbit, the population will be approximately **599,210 rabbits** after another 66 months.

Would you like more details on this calculation or have any other questions?

Here are some related questions to consider:
1. What would the population be after 20 years?
2. How does the growth rate change if the population grows to 500,000 in 11 years instead of 345,345?
3. How long would it take for the rabbit population to reach 1 million rabbits?
4. What factors could cause the actual population growth to deviate from this exponential model?
5. How would the presence of predators affect the growth rate of the rabbit population?

**Tip:** When working with exponential growth models, always make sure the time units are consistent throughout the calculation.

Learn how to calculate rabbit population growth using the exponential growth model. Start with an initial population of 115,115 rabbits and project the population after 16.5 years. Includes step-by-step calculation using natural logarithms and exponential functions.

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