To determine the future value of Isabella's investment using the formula for continuous compounding, we can apply the formula \( A = Pe^{rt} \).
Where:
- \( A \) = the amount of money accumulated after n years, including interest.
- \( P \) = the principal amount (the initial amount of money).
- \( r \) = the annual interest rate (decimal).
- \( t \) = the time the money is invested or borrowed for, in years.
- \( e \) = the base of the natural logarithm, approximately equal to 2.71828.
Given:
- \( P = 10,000 \)
- \( r = 0.03 \) (3% as a decimal)
- \( t = 15 \)
Now, we plug the values into the formula:
\[ A = 10,000 \cdot e^{(0.03 \cdot 15)} \]
Calculating \( 0.03 \cdot 15 \):
\[ 0.03 \cdot 15 = 0.45 \]
Now we calculate \( e^{0.45} \):
Using a calculator or mathematical software, we find:
\[ e^{0.45} \approx 1.56831 \]
Now, we can substitute this value back into the formula:
\[ A \approx 10,000 \cdot 1.56831 \approx 15,683.10 \]
Therefore, Isabella’s investment will be worth approximately $15,683.10 in 15 years.
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