Question

Kevin opened a savings account 1 year ago. The account earns 7% interest, compounded continuously. If the current balance is $100.00, how much did he deposit initially?
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 initial deposit \( P \) that Kevin made, we will use the formula for continuously compounded interest:

\[
A = Pe^{rt}
\]

Where:
- \( A \) is the final amount (current balance) = $100.00
- \( P \) is the principal (initial deposit we need to find)
- \( r \) is the interest rate expressed as a decimal = 7% = 0.07
- \( t \) is the time in years = 1 year
- \( e \) is the base of natural logarithms approximately equal to 2.71828

We can rearrange the formula to solve for \( P \):

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

Now, substituting the values we have:

\[
P = \frac{100}{e^{0.07 \cdot 1}} = \frac{100}{e^{0.07}}
\]

Next, we calculate \( e^{0.07} \):

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

Now, we can calculate \( P \):

\[
P = \frac{100}{1.072508} \approx 93.24
\]

Thus, Kevin's initial deposit was approximately:

\[
\boxed{93.24}
\]