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.