https://file.aiquickdraw.com/mathbot/file/1725916414451r395umry.jpg

Let's break down the exponential growth problem shown in the image step-by-step.

### Problem:
The rabbit population on a game reserve doubles every 3 months. Initially, there were 130 rabbits. The population can be modeled by the exponential function:
\[
P(t) = P_0 \cdot a^t
\]
where:
- \(P_0 = 130\) (initial population),
- \(a\) is the growth rate constant (to be determined),
- \(t\) is time in months.

### Part (a) - Determine the growth rate constant \(a\):
The population doubles every 3 months, so after 3 months:
\[
P(3) = 2 \cdot P_0
\]
Using the exponential function, this gives:
\[
2 \cdot P_0 = P_0 \cdot a^3
\]
Cancel \(P_0\) from both sides:
\[
2 = a^3
\]
Solving for \(a\):
\[
a = \sqrt[3]{2} \approx 1.2599
\]
So, the growth rate constant \(a\) is approximately **1.2599** (rounded to 4 decimal places).

### Part (b) - Find the function for the rabbit population after \(t\) months:
Now that we know \(a \approx 1.2599\), substitute this value along with \(P_0 = 130\) into the exponential formula:
\[
P(t) = 130 \cdot 1.2599^t
\]

### Part (c) - Determine the population after 13 months:
To find \(P(13)\), substitute \(t = 13\) into the equation from part (b):
\[
P(13) = 130 \cdot 1.2599^{13}
\]
Let me calculate this for you.

### Part (d) - Select the graph of the function:
Once we calculate \(P(13)\), we can check which graph best matches the exponential growth pattern.

I'll now compute \(P(13)\).After 13 months, the rabbit population is approximately **2620** rabbits (rounded to the nearest whole number).

### Summary of answers:
1. The growth rate constant \( a \) is approximately **1.2599**.
2. The population function is \( P(t) = 130 \cdot 1.2599^t \).
3. The population after 13 months is approximately **2620** rabbits.
4. Now, you can select the graph that best represents this exponential growth.

Would you like more details or have any questions?

Here are some additional related questions:
1. How would the function change if the population tripled every 3 months?
2. What would the population be after 24 months?
3. How does changing the initial population affect the growth curve?
4. What is the doubling time if the population grows continuously instead of every 3 months?
5. How does exponential growth differ from linear growth in terms of long-term population size?

**Tip:** Exponential functions grow rapidly, so small changes in the growth rate \(a\) can result in very large differences in population over time.

The rabbit population doubles every 3 months. Suppose there were 130 rabbits initially. The population can be modeled by the exponential function P(t) = P0(a)^t, where t is the number of months, P0 is the initial population, and a is the growth rate constant.

Solve the exponential growth problem involving a rabbit population doubling every 3 months with an initial population of 130. Use the formula P(t) = P0 * a^t to determine the population after a given time, the growth constant, and select the graph that best represents the population growth.

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