Question

Elise earned some money doing odd jobs last summer and put it in a savings account that earns 1% interest compounded continuously. After 4 years, there is $500.00 in the account. How much did Elise earn doing odd jobs?
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 how much Elise earned doing odd jobs (the principal \( P \)), we can use the formula for continuous compounding:

\[
A = Pe^{rt}
\]

Where:
- \( A \) is the total amount after time \( t \) (which is $500.00),
- \( P \) is the principal amount (what we want to find),
- \( e \) is approximately 2.71828,
- \( r \) is the interest rate (1% or 0.01 in decimal form),
- \( t \) is the time in years (which is 4 years).

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

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

Now we can plug in the values:

\[
A = 500, \quad r = 0.01, \quad t = 4
\]

First, we need to calculate \( e^{rt} \):

\[
rt = 0.01 \times 4 = 0.04
\]

Now calculate \( e^{0.04} \):

\[
e^{0.04} \approx 1.04081 \quad (\text{using a calculator})
\]

Now we can substitute this back into the equation for \( P \):

\[
P = \frac{500}{1.04081}
\]

Calculating this gives:

\[
P \approx \frac{500}{1.04081} \approx 480.43
\]

Thus, Elise earned approximately **$480.43** doing odd jobs, rounded to the nearest cent.