Find the interest earned on a $14,000 balance for nine months at 1.1% interest compounded continuously.

To calculate the interest earned on a balance with **continuous compounding**, we use the formula:

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

Where:
- $ A $ is the amount after time $ t $ (the total balance including interest).
- $ P $ is the principal amount (initial balance).
- $ r $ is the annual interest rate (in decimal form).
- $ t $ is the time in years.
- $ e $ is the base of the natural logarithm (approximately 2.71828).

In this case:
- $ P = 14,000 $ (the principal).
- $ r = 0.011 $ (1.1% as a decimal).
- $ t = \frac{9}{12} = 0.75 $ years (since nine months is 0.75 of a year).

Let's calculate the total amount $ A $ and then subtract the principal $ P $ to find the interest earned.

\[
A = 14,000 \times e^{(0.011 \times 0.75)}
\]

Now, calculating the interest.The interest earned on a $14,000 balance for nine months at a 1.1% interest rate compounded continuously is approximately **$115.98**.

Would you like more details or have any questions? Here are some related questions to explore further:

1. How does continuous compounding differ from simple or compound interest?
2. What is the formula for compound interest when compounded annually or monthly?
3. How would the interest change if the rate was 2% instead of 1.1%?
4. What would be the total amount (principal + interest) after one year instead of nine months?
5. How does the time period affect the interest earned with continuous compounding?

**Tip:** Continuous compounding maximizes interest growth compared to other compounding methods due to its constant accumulation of interest.

Learn how to calculate the interest earned on a $14,000 balance at a 1.1% annual interest rate compounded continuously for nine months. Using the formula A = P * e^(rt), this solution demonstrates key concepts of continuous compounding and exponential growth, suitable for students in Grades 10-12.

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