To solve for the monthly growth rate of the population, we use the exponential growth formula:

$P(t) = P_0 \cdot e^{r \cdot t}$

Where:

• $P(t)$ is the population at time $t$,
• $P_0$ is the initial population,
• $r$ is the monthly growth rate,
• $t$ is the time in months,
• $e$ is Euler's number (approximately 2.71828).

We are given:

• $P_0 = 1200$ (initial population),
• $P(t) = 5000$ (population after 24 months),
• $t = 24$ months.

Step 1: Plug the known values into the formula:

$5000 = 1200 \cdot e^{r \cdot 24}$

Step 2: Solve for $e^{r \cdot 24}$:

$\frac{5000}{1200} = e^{24r}$ $\frac{5000}{1200} = 4.1667$

$e^{24r} = 4.1667$

Step 3: Take the natural logarithm (ln) of both sides:

$\ln(4.1667) = 24r$ $1.427 = 24r$

Step 4: Solve for $r$:

$r = \frac{1.427}{24} = 0.0595$

Step 5: Convert $r$ to a percentage:

$r \times 100 = 5.95\%$

Final Answer:

The monthly growth rate is 5.95%.

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Let me know if you'd like further details or have any questions. Here's some additional related content:

1. How do you use the exponential decay formula?
2. How would the formula change if the population were decreasing?
3. How can you estimate future populations with different time intervals?
4. What factors could influence the growth rate in a real-world scenario?
5. How does the time variable impact the population model in terms of rates of change?

Tip: When solving exponential growth problems, always ensure the time unit (months, years, etc.) is consistent with the time variable in your formula.