The population of rabbits on an island is growing exponentially. In the year 2005, the population of rabbits was 6900, and by 2012 the population had grown to 13500. Predict the population of rabbits in the year 2015, to the nearest whole number.

To solve this problem, we can 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.
- \( P_0 = 6900 \) (population in 2005).
- \( P(7) = 13500 \) (population in 2012, which is 7 years after 2005).
- \( t = 7 \) (time in years from 2005 to 2012).

### Step 2: Set up the equation for 2012 and solve for the growth rate \( r \).

\[
13500 = 6900 \cdot e^{7r}
\]

Divide both sides by 6900:

\[
\frac{13500}{6900} = e^{7r}
\]

Simplify:

\[
\frac{13500}{6900} = \frac{135}{69} \approx 1.9565
\]

Now, take the natural logarithm (ln) of both sides:

\[
\ln(1.9565) = 7r
\]

\[
r \approx \frac{\ln(1.9565)}{7}
\]

### Step 3: Calculate \( r \).

\[
r \approx \frac{0.6728}{7} \approx 0.0961
\]

### Step 4: Predict the population in 2015 (10 years after 2005).

Now, calculate \( P(10) \) using the exponential growth formula:

\[
P(10) = 6900 \cdot e^{0.0961 \cdot 10}
\]

\[
P(10) = 6900 \cdot e^{0.961}
\]

### Step 5: Compute \( P(10) \).

\[
P(10) \approx 6900 \cdot 2.6146 \approx 18040
\]

### Final Answer:
The population of rabbits in the year 2015 is approximately **18,040**.

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

Here are 5 related questions to further your understanding:
1. How would the population change if the growth rate was higher?
2. What would the population be in 2020 using the same growth rate?
3. How can exponential decay be modeled similarly?
4. What happens to the population if the growth rate is negative?
5. How would the model change if there was a carrying capacity on the island?

**Tip:** Always check the units and context when interpreting the growth rate in exponential models.

Learn how to predict the population growth of rabbits on an island using the exponential growth formula. Given initial and future populations, calculate growth rate and predict future populations. Suitable for Grades 9-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation