Question

In 1 year, Quinn wants to buy a bicycle that costs $500.00. If he opens a savings account that earns 5% interest compounded continuously, how much will he have to deposit as principal to have enough money in 1 year to buy 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 find the principal \( P \) that Quinn needs to deposit in order to have $500 in 1 year, we'll use the continuous compounding interest formula given by:

\[
A = Pe^{rt}
\]

Where:
- \( A \) is the final amount (in this case, $500),
- \( P \) is the principal,
- \( r \) is the interest rate (5%, or 0.05 as a decimal),
- \( t \) is the time in years (1 year).

We need to solve for \( P \):

\[
500 = Pe^{0.05 \cdot 1}
\]

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

Using a calculator, we find:

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

Substituting this back into the equation, we get:

\[
500 = P \cdot 1.051271
\]

Now, we isolate \( P \):

\[
P = \frac{500}{1.051271}
\]

Calculating this gives:

\[
P \approx \frac{500}{1.051271} \approx 476.188
\]

Rounding to the nearest cent, we find:

\[
P \approx 476.19
\]

Thus, Quinn will need to deposit approximately **$476.19** as principal to have enough money to buy the bicycle in 1 year.