The rabbit population on an island starts off as 120 and it grows by 15% per year. How long will it take for the rabbit population to reach 5,000?

This problem involves exponential growth, where the population grows by a fixed percentage each year. The formula for exponential growth is:

\[
P(t) = P_0 \times (1 + r)^t
\]

Where:
- \(P(t)\) is the population after time \(t\),
- \(P_0\) is the initial population (120 in this case),
- \(r\) is the growth rate per year (15% or 0.15),
- \(t\) is the time in years.

We are looking for \(t\) when \(P(t) = 5000\), so:

\[
5000 = 120 \times (1 + 0.15)^t
\]

First, divide both sides by 120:

\[
\frac{5000}{120} = (1.15)^t
\]

\[
41.67 = (1.15)^t
\]

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

\[
\ln(41.67) = t \times \ln(1.15)
\]

Solving for \(t\):

\[
t = \frac{\ln(41.67)}{\ln(1.15)}
\]

Let me calculate this value for \(t\).It will take approximately 26.69 years for the rabbit population to reach 5,000.

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

Here are 5 related questions for further exploration:
1. How would the time change if the growth rate were 10% instead of 15%?
2. What if the population goal was 10,000 instead of 5,000?
3. How does the formula change if the growth is compounded monthly instead of yearly?
4. What would the population be after 10 years at the same growth rate?
5. How long would it take for the population to double?

**Tip:** Using logarithms is crucial when solving for time in exponential growth problems.

Find out how long it takes for a rabbit population to grow from 120 to 5,000 with a 15% annual increase. This problem covers exponential growth and logarithms, ideal for students in Grades 10-12. Learn to use the exponential growth formula and logarithms to solve this real-world problem.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation