 If $5000 is invested at 5.25% compounded continuously, what is the amount in the account after 10
years?

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.

If $5000 is invested at 5.25% compounded continuously, what is the amount in the account after 10 years?

Learn how to calculate the future value of $5000 invested at a continuous compounding interest rate of 5.25% for 10 years. This example covers the formula A = P * e^(rt) and exponential growth theory, demonstrating how the final amount is approximately $8453.

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