To solve this problem, we use the formula for continuous compounding:

$A = P e^{rt}$

Where:

• $A$ is the amount in the account after time $t$,
• $P$ is the initial principal (in this case, $5000),
• $r$ is the annual interest rate (as a decimal),
• $t$ is the time in years,
• $e$ is Euler's number (approximately 2.71828).

Given:

• $P = 5000$,
• $r = 5.25\%$ or $r = 0.0525$,
• $t = 10$ years.

Now we substitute these values into the formula:

$A = 5000 \times e^{0.0525 \times 10}$

Simplifying the exponent:

$A = 5000 \times e^{0.525}$

Using $e^{0.525} \approx 1.6906$:

$A = 5000 \times 1.6906 = 8453$

So, the amount in the account after 10 years is approximately $8453.

Would you like to go over any details of this solution?

Here are 5 related questions to expand on this concept:

1. What happens if the interest rate is compounded monthly instead of continuously?
2. How would the final amount change if the investment period was 15 years?
3. What is the difference between continuous compounding and compounding quarterly?
4. How do you find the doubling time of an investment using continuous compounding?
5. What if the interest rate was 6% instead of 5.25%? How would the outcome change?

Tip: Continuous compounding yields the maximum possible interest for a given rate, because it compounds at every instant.