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.