Question

Colton and Pam deposit $5,000.00 into a savings account which earns 5% interest compounded continuously. They want to use the money in the account to go on a trip in 1 year. How much will they be able to spend?
Round your answer to the nearest cent.

Answer 1

To calculate the amount of money in a continuously compounded interest account, you can use the formula:

\[
A = Pe^{rt}
\]

Where:
- \( A \) = the amount of money accumulated after \( t \) years, including interest.
- \( P \) = the principal amount (the initial deposit or investment).
- \( r \) = the annual interest rate (decimal).
- \( t \) = the number of years the money is invested or borrowed.
- \( e \) = the base of the natural logarithm (approximately equal to 2.71828).

For Colton and Pam:
- \( P = 5000 \)
- \( r = 0.05 \)
- \( t = 1 \)

Plugging these values into the formula gives:

\[
A = 5000 \cdot e^{0.05 \times 1}
\]

First, calculate the exponent:

\[
0.05 \times 1 = 0.05
\]

Now, calculate \( e^{0.05} \):

\[
e^{0.05} \approx 1.051271
\]

Now, substitute that back into the formula:

\[
A = 5000 \cdot 1.051271
\]
\[
A \approx 5256.36
\]

Rounding to the nearest cent, Colton and Pam will be able to spend approximately **$5,256.36**.