Question

In 1 year, Mandy wants to buy a bicycle that costs $800.00. If she opens a savings account that earns 15% interest compounded continuously, how much will she have to deposit as principal to have enough money in 1 year to buy the bike?
Round your answer to the nearest cent.
$

Answer 1

To find out how much Mandy needs to deposit as principal in a savings account that earns continuous interest to reach a goal of $800 in one year, we can use the formula for continuous compounding:

\[
A = Pe^{rt}
\]

Where:
- \( A \) is the amount of money accumulated after n years, including interest ($800 in this case),
- \( P \) 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 2.71828).

We need to solve for \( P \):

\[
800 = Pe^{0.15 \times 1}
\]

Calculating \( e^{0.15} \):

\[
e^{0.15} \approx 1.161834
\]

Now, plug this value back into the equation:

\[
800 = P \times 1.161834
\]

Now, solve for \( P \):

\[
P = \frac{800}{1.161834}
\]

Calculating the value of \( P \):

\[
P \approx \frac{800}{1.161834} \approx 688.29
\]

Thus, rounding to the nearest cent, Mandy needs to deposit approximately **$688.29**.