We are given the exponential growth model for the population of bunnies on Grassygreen Island:

$B(t) = P e^{rt}$

Where:

• $P$ is the initial population.
• $r$ is the growth rate (1.4% per day or 0.014 as a decimal).
• $t$ is the time in days.

Part A: Finding the population 30 days from now

We know:

• $P = 179$ (initial population),
• $r = 0.014$,
• $t = 30$.

Substitute these values into the exponential growth formula:

$B(30) = 179 e^{0.014 \times 30}$

Now let's calculate this.

Part B: How long it will take for the population to reach 4000?

We are given:

• $B(t) = 4000$,
• $P = 179$,
• $r = 0.014$.

We need to find $t$ such that:

$4000 = 179 e^{0.014t}$

We can solve this by isolating $t$:

1. Divide both sides by 179: $\frac{4000}{179} = e^{0.014t}$

2. Take the natural logarithm (ln) of both sides to eliminate the exponential: $\ln\left(\frac{4000}{179}\right) = 0.014t$

3. Finally, solve for $t$: $t = \frac{\ln\left(\frac{4000}{179}\right)}{0.014}$

Let me calculate both parts for you.### Part A: The population 30 days from now will be approximately 272 bunnies.

Part B:

It will take approximately 222 days for the population to reach 4000 bunnies.

Do you need more details or have any questions?

Here are five related questions to explore further:

1. What would the population be after 60 days?
2. How would a different growth rate, say 2% per day, affect the time to reach 4000?
3. What is the doubling time for the bunny population with the current growth rate?
4. How does the exponential growth model compare with a linear growth model?
5. How would the population change if the growth rate decreased over time?

Tip: Exponential growth can rapidly escalate over time, especially with even small percentage increases.