To solve this problem using the continuous compound interest formula, we use the following formula:

$A = P e^{rt}$

Where:

• $A$ is the final amount ($22,730)
• $P$ is the principal amount ($15,000)
• $r$ is the interest rate (which we need to find)
• $t$ is the time in years (convert months to years)

Step 1: Convert the time from months to years: $t = \frac{60}{12} = 5 \text{ years}$

Step 2: Substitute the known values into the formula: $22,730 = 15,000 \cdot e^{r \cdot 5}$

Step 3: Divide both sides by 15,000: $\frac{22,730}{15,000} = e^{5r}$ $1.51533 = e^{5r}$

Step 4: Take the natural logarithm (ln) of both sides to get rid of the exponential: $\ln(1.51533) = 5r$ $0.4146 = 5r$

Step 5: Solve for $r$: $r = \frac{0.4146}{5} = 0.08292$

Step 6: Convert to percentage: $r = 8.292\%$

Thus, the interest rate is approximately 8.292%.

Do you want more details or have any questions?

Here are 5 related questions:

1. How is the continuous compound interest formula derived?
2. What is the difference between continuous compounding and simple interest?
3. Can you solve the problem if the time period is given in days?
4. How does continuous compounding affect the growth of investments compared to regular compounding?
5. How can you calculate future value with different compounding intervals?

Tip: Always ensure to convert time to years when using the continuous compound interest formula, as $r$ is expressed annually.