Question

In 1 year, Duncan wants to buy a bicycle that costs $400.00. If he opens a savings account that earns 4% 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 determine how much Duncan needs to deposit as principal (P) in order to accumulate $400 in one year with an interest rate of 4% compounded continuously, we use the formula for continuous compounding interest:

\[ A = Pe^{rt} \]

where:
- \( A = 400 \) (the amount he wants to have),
- \( r = 0.04 \) (the interest rate as a decimal),
- \( t = 1 \) (the time in years),
- \( e \) is approximately \( 2.71828 \).

We need to rearrange this formula to solve for \( P \):

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

Substituting the known values:

\[
P = \frac{400}{e^{(0.04)(1)}}
\]

Calculating \( e^{0.04} \):

\[
e^{0.04} \approx 1.04081
\]

Now, substituting this back into the equation for \( P \):

\[
P \approx \frac{400}{1.04081}
\]

Calculating \( P \):

\[
P \approx 384.17
\]

Therefore, the amount of principal Duncan needs to deposit is approximately **$384.17**. When rounded to the nearest cent, the answer is:

\[
\boxed{384.17}
\]