Question

Chase opened a savings account and deposited $300.00. The account earns 7% interest, compounded continuously. If he wants to use the money to buy a new bicycle in 1 year, how much will he be able to spend on the bike?
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 how much money Chase will have after 1 year with continuous compounding interest, we can use the formula:

\[
A = P e^{rt}
\]

Where:
- \( A \) is the balance (final amount),
- \( P \) is the principal (starting amount),
- \( e \) is the base of natural logarithms (approximately 2.71828),
- \( r \) is the annual interest rate (as a decimal),
- \( t \) is the time in years.

Given:
- \( P = 300 \)
- \( r = 0.07 \) (7% expressed as a decimal)
- \( t = 1 \)

Now, substituting the values into the formula:

\[
A = 300 \times e^{0.07 \times 1}
\]

Calculating the exponent:

\[
0.07 \times 1 = 0.07
\]

Now we will calculate \( e^{0.07} \):

\[
e^{0.07} \approx 2.71828^{0.07} \approx 1.072508
\]

Now substituting this value back into the equation for \( A \):

\[
A \approx 300 \times 1.072508 \approx 321.7524
\]

Rounding to the nearest cent:

\[
A \approx 321.75
\]

Therefore, Chase will be able to spend approximately **$321.75** on the bike.