Question

Isabella invests 10,000 in an account that is compounded continuously at an annual interest rate of 3%. According to the formula A = Pe^rt. Apply the formula to find out how much Isabella’s investment will be worth in 15 years

Answer 1

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.