Question

Reid and Diana deposit $700.00 into a savings account which earns 3% 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?
Use the formula A=Pert, where A is the balance (final amount), P is the principal (starting amount), e is the base of natural logarithms (≈2.71828), r is the interest rate expressed as a decimal, and t is the time in years.
Round your answer to the nearest cent.

Answer 1

To calculate the future amount in the savings account after one year with continuous compounding, we will use the formula:

\[
A = Pe^{rt}
\]

where:
- \( P \) is the principal amount (initial deposit)
- \( r \) is the interest rate (as a decimal)
- \( t \) is the time in years
- \( e \) is the base of natural logarithms (approximately 2.71828)

Given:
- \( P = 700.00 \)
- \( r = 0.03 \) (3% interest rate)
- \( t = 1 \) year

Now we can substitute the values into the formula:

\[
A = 700.00 \cdot e^{0.03 \cdot 1}
\]

Calculating the exponent:

\[
0.03 \cdot 1 = 0.03
\]

Now we calculate \( e^{0.03} \):

Using \( e \approx 2.71828 \):

\[
e^{0.03} \approx 2.71828^{0.03} \approx 1.03045453395
\]

Now substituting back into the formula:

\[
A = 700.00 \cdot 1.03045453395 \approx 721.318173765
\]

Now rounding to the nearest cent:

\[
A \approx 721.32
\]

So, Reid and Diana will be able to spend **$721.32** for their trip.