If you deposit $2,500 into an account that has an annual rate of 6.3% compounded continuously, then how much interest is earned after 10 months?
Round your answer to two decimal places. Do not include the $ sign.
Your Answer:

To calculate the interest earned on an account with continuous compounding, we use the formula:

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

Where:
- $A$ is the final amount,
- $P$ is the principal amount (\$2,500),
- $r$ is the annual interest rate (6.3% or 0.063),
- $t$ is the time in years (10 months = $ \frac{10}{12} = 0.8333$ years),
- $e$ is the mathematical constant approximately equal to 2.71828.

### Step 1: Plug in the values
\[
A = 2500 \cdot e^{0.063 \cdot 0.8333}
\]

### Step 2: Calculate the exponent
\[
0.063 \cdot 0.8333 = 0.0525
\]

\[
e^{0.0525} \approx 1.0539
\]

### Step 3: Calculate the final amount
\[
A = 2500 \cdot 1.0539 \approx 2634.75
\]

### Step 4: Subtract the principal to find the interest earned
\[
\text{Interest} = A - P = 2634.75 - 2500 = 134.75
\]

### Final Answer:
The interest earned after 10 months is **134.75**.

Would you like more details or have any questions?  

Here are 5 related questions to expand your understanding:
1. How does continuous compounding differ from annual or monthly compounding?
2. What is the formula for compound interest if it's compounded annually instead?
3. What would the interest be if the time was 2 years instead of 10 months?
4. How does the interest change if the annual rate is increased to 8%?
5. What is the effect of extending the compounding period to 20 months instead of 10?

**Tip:** Always verify the compounding method (continuous, monthly, yearly, etc.) when solving financial problems, as it can significantly affect the result.

If you deposit $2,500 into an account that has an annual rate of 6.3% compounded continuously, then how much interest is earned after 10 months? Round your answer to two decimal places. Do not include the $ sign.

Learn how to calculate interest earned on a $2,500 deposit with a 6.3% annual rate compounded continuously over 10 months. This detailed solution includes step-by-step calculations using the formula A = P * e^(rt). Ideal for students in Grades 10-12 studying financial mathematics.

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