To solve this problem, we’ll use the continuous compound interest formula:

$A = P e^{rt}$

where:

• $A$ is the amount of money accumulated after time $t$,
• $P$ is the initial principal,
• $r$ is the interest rate,
• $t$ is the time in years, and
• $e$ is the base of the natural logarithm, approximately equal to 2.718.

We are given:

• $A = 22141$,
• $P = 13800$,
• $t = 60$ months (which we’ll convert to years: $t = 60/12 = 5$ years),
• $r$ is unknown.

Steps to Solve for $r$

1. Substitute the known values into the formula: $22141 = 13800 \cdot e^{5r}$

2. Divide both sides by 13800: $\frac{22141}{13800} = e^{5r}$

3. Take the natural logarithm (ln) of both sides to solve for $r$: $\ln\left(\frac{22141}{13800}\right) = 5r$

4. Solve for $r$ by dividing both sides by 5: $r = \frac{\ln\left(\frac{22141}{13800}\right)}{5}$

Let’s calculate the value.The interest rate $r$ is approximately $0.0946$, or 9.46% per year when compounded continuously.

Would you like further details on this solution, or have any additional questions?

Here are five related questions you might consider:

1. How would the solution change if the interest were compounded monthly instead of continuously?
2. What is the formula for compound interest if it’s compounded $n$ times per year?
3. How does continuous compounding differ from simple compounding in terms of growth rate?
4. If the interest rate $r$ were doubled, what would be the new accumulated amount after 5 years?
5. How can we use logarithmic properties to solve for time $t$ if $r$ and $A$ are known?

Tip: Continuous compounding is ideal for scenarios where investments are growing constantly, like certain financial or biological growth models.