KEEP 10 DECIMAL PLACES WHEN PERFORMING CALCULATIONS

The population of rabbits on an island is growing exponentially. In the year 2000, the population of rabbits was 950, and by 2003 the population had grown to 1000. Predict the population of rabbits in the year 2013, *to the nearest whole number*.

To solve this problem with high precision, we will use the exponential growth formula:

\[
P(t) = P_0 \cdot e^{rt}
\]

Where:
- \( P(t) \) is the population at time \( t \).
- \( P_0 \) is the initial population.
- \( r \) is the growth rate.
- \( t \) is the time elapsed.

### Step 1: Identify the known values.
- \( P_0 = 950 \) (population in 2000).
- \( P(3) = 1000 \) (population in 2003, which is 3 years after 2000).
- \( t = 3 \) (time in years from 2000 to 2003).

### Step 2: Set up the equation for 2003 and solve for the growth rate \( r \).

\[
1000 = 950 \cdot e^{3r}
\]

Divide both sides by 950:

\[
\frac{1000}{950} = e^{3r}
\]

Simplify:

\[
\frac{1000}{950} \approx 1.0526315789
\]

Take the natural logarithm (ln) of both sides:

\[
\ln(1.0526315789) = 3r
\]

Solve for \( r \):

\[
r \approx \frac{\ln(1.0526315789)}{3}
\]

### Step 3: Calculate \( r \) using 10 decimal places.

\[
r \approx \frac{0.0512932944}{3} \approx 0.0170977648
\]

### Step 4: Predict the population in 2013 (13 years after 2000).

Now, calculate \( P(13) \) using the exponential growth formula:

\[
P(13) = 950 \cdot e^{0.0170977648 \cdot 13}
\]

\[
P(13) = 950 \cdot e^{0.2222709424}
\]

### Step 5: Compute \( P(13) \).

\[
P(13) \approx 950 \cdot 1.2489512488 \approx 1186.50368638
\]

### Final Answer:
The population of rabbits in the year 2013 is approximately **1187** (to the nearest whole number).

Would you like more details, or do you have any questions?

Here are 5 related questions to further your understanding:
1. What would the population be in 2015 using the same growth rate?
2. How would the population change if the growth rate was lower?
3. How does the model change if the population were to double in 3 years?
4. How would this model apply if there were a limit to the population's growth?
5. What is the importance of using precise decimal places in growth calculations?

**Tip:** Maintaining precision in intermediate steps is crucial in mathematical modeling, especially when the final results are sensitive to small changes in the parameters.

Learn how to predict the population growth of rabbits using the exponential growth formula. Given initial and future population data, calculate the population in specific years with precision. Explore the mathematical concepts behind exponential growth and apply them to real-world scenarios.

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