Question

Ayana opened a savings account and deposited $5,000.00 as principal. The account earns 4% interest, compounded continuously. What is the balance after 4 years?
Round your answer to the nearest cent.
$

Answer 1

To calculate the balance of a continuously compounded interest account, we use the formula:

\[
A = Pe^{rt}
\]

where:
- \( A \) is the amount of money accumulated after n years, including interest.
- \( P \) is the principal amount (the initial amount of money).
- \( r \) is the annual interest rate (decimal).
- \( t \) is the time the money is invested for in years.
- \( e \) is the base of the natural logarithm, approximately equal to 2.71828.

Given:
- \( P = 5000 \)
- \( r = 0.04 \) (4% as a decimal)
- \( t = 4 \)

Now we plug these values into the formula:

\[
A = 5000 \times e^{0.04 \times 4}
\]

First, calculate \( 0.04 \times 4 = 0.16 \).

Next, calculate \( e^{0.16} \):

\[
e^{0.16} \approx 1.17351
\]

Now multiply by the principal:

\[
A \approx 5000 \times 1.17351 \approx 5867.55
\]

Thus, the balance after 4 years, rounded to the nearest cent, is:

\[
\boxed{5867.55}
\]