A total of $25,600 is invested at an annual interest rate of 6%. Find the balance in the account
after 15 years if it is compounded continuously (round to the nearest dollar).
$62,966
$62,956
$62,866
$63,166

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

\[
A = P e^{rt}
\]

where:
- $ A $ is the amount of money accumulated after $ t $ years, including interest.
- $ P $ is the principal amount (the initial investment).
- $ r $ is the annual interest rate (in decimal form).
- $ t $ is the time the money is invested or borrowed for, in years.
- $ e $ is Euler's number (approximately $ 2.71828 $).

Given:
- $ P = 25,600 $
- $ r = 6\% = 0.06 $
- $ t = 15 $ years

Now, we plug these values into the formula:

\[
A = 25,600 \times e^{0.06 \times 15}
\]

Let's calculate the value.

The balance in the account after 15 years, compounded continuously at an annual interest rate of 6%, is **$62,966**.

Would you like more details or have any questions?

Here are 5 related questions to expand on this concept:
1. What would the balance be after 10 years with the same interest rate?
2. How does continuous compounding differ from annual compounding?
3. How much interest would be earned after 15 years on the initial investment?
4. What happens if the interest rate increases to 8%?
5. How would the balance change if the investment was compounded quarterly instead of continuously?

**Tip:** Continuous compounding grows faster than any other form of compounding because interest is added at every moment.

A total of $25,600 is invested at an annual interest rate of 6%. Find the balance in the account after 15 years if it is compounded continuously (round to the nearest dollar).

Calculate the balance of a $25,600 investment after 15 years with a 6% interest rate using the continuous compounding formula. This detailed solution covers exponential functions and continuous compounding, making it suitable for Grades 10-12 students.

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