Find the amount in a continuously compounded account for the following condition.

Principal, $5000; Annual interest rate, 5.6%; time, 3 years.
The balance after 3 years is $____. (Round the final answer to the nearest cent as needed. Round all intermediate values to five decimal places as needed.)

1 answer

To find the amount in a continuously compounded account, you can use the formula:

\[ A = Pe^{rt} \]

where:

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

Given:

  • \( P = 5000 \)
  • \( r = 5.6% = 0.056 \)
  • \( t = 3 \)

Plugging in the values:

\[ A = 5000 \times e^{(0.056 \times 3)} \]

Calculating the exponent:

\[ 0.056 \times 3 = 0.168 \]

Now, calculating \( e^{0.168} \):

\[ e^{0.168} \approx 1.18380 \] (rounded to five decimal places)

Now, we can calculate \( A \):

\[ A = 5000 \times 1.18380 \]

\[ A \approx 5919.00 \]

Therefore, the balance after 3 years is approximately:

\[ \boxed{5919.00} \]

(rounded to the nearest cent).