Question

In 1 year, Richard wants to buy a bicycle that costs $1,000.00. If he opens a savings account that earns 1% interest compounded continuously, how much will he 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 Richard needs to deposit as principal to have $1,000 in one year with an interest rate of 1% compounded continuously, we can use the formula for continuous compounding, which is given by:

\[
A = Pe^{rt}
\]

where:
- \(A\) is the amount of money accumulated after time \(t\),
- \(P\) 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 equal to 2.71828).

In this case:
- \(A = 1000\),
- \(r = 0.01\) (1% as a decimal),
- \(t = 1\).

We need to solve for \(P\). Rearranging the formula gives:

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

Substituting in the values:

\[
P = \frac{1000}{e^{0.01 \times 1}}
\]
\[
P = \frac{1000}{e^{0.01}}
\]

Calculating \(e^{0.01}\):

\[
e^{0.01} \approx 1.01005016708
\]

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

\[
P = \frac{1000}{1.01005016708} \approx 990.049
\]

Rounding to the nearest cent, Richard will need to deposit:

\[
\boxed{990.05}
\]