Question

Alana opened a savings account 1 year ago. The account earns 14% interest, compounded continuously. If the current balance is $200.00, how much did she deposit initially?
Round your answer to the nearest cent.
$

Answer 1

To find the initial deposit Alana made in her savings account, we can use the formula for continuous compounding interest, 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 deposit),
- \( r \) is the annual interest rate (decimal),
- \( t \) is the time the money is invested or borrowed for, in years,
- \( e \) is Euler's number, approximately equal to 2.71828.

In this case:
- \( A = 200.00 \) (the current balance),
- \( r = 0.14 \) (14% interest rate as a decimal),
- \( t = 1 \) year.

We need to solve for \( P \):

\[
200.00 = Pe^{0.14 \cdot 1}
\]
\[
200.00 = Pe^{0.14}
\]

Now we can isolate \( P \):

\[
P = \frac{200.00}{e^{0.14}}
\]

Calculating \( e^{0.14} \):

\[
e^{0.14} \approx 1.150274
\]

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

\[
P = \frac{200.00}{1.150274} \approx 173.48
\]

Rounding to the nearest cent:

\[
P \approx 173.48
\]

Thus, the initial deposit Alana made was approximately **$173.48**.