To calculate the future value \( A \) of an investment compounded continuously, you can use the formula:
\[ A = Pe^{rt} \]
Where:
- \( P \) is the principal (initial investment),
- \( r \) is the annual interest rate (as a decimal),
- \( t \) is the time in years, and
- \( e \) is Euler's number, approximately equal to 2.71828.
Given:
- \( P = 10,000 \)
- \( r = 0.03 \) (3% as a decimal)
- \( t = 15 \)
Now, substitute the values into the formula:
\[ A = 10,000 \times e^{0.03 \times 15} \]
Calculate \( rt \):
\[ rt = 0.03 \times 15 = 0.45 \]
Now substitute this value back into the formula:
\[ A = 10,000 \times e^{0.45} \]
Next, calculate \( e^{0.45} \):
Using a calculator:
\[ e^{0.45} \approx 1.5708 \]
Now multiply by the principal:
\[ A \approx 10,000 \times 1.5708 \approx 15,708 \]
Rounding to the nearest dollar, Isabella’s investment will be worth approximately $15,708.
Based on the options provided, the closest response would be $15,683.
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