Question

Troy earned some money doing odd jobs last summer and put it in a savings account that earns 4% interest compounded continuously. After 7 years, there is $600.00 in the account. How much did Troy 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 solve for the principal amount \( P \) that Troy initially earned, we can use the formula for continuously compounded interest:

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

Where:
- \( A \) is the final amount (which is $600.00),
- \( P \) is the principal amount we want to find,
- \( r \) is the interest rate (4%, or 0.04 as a decimal),
- \( t \) is the time in years (7 years),
- \( e \) is approximately 2.71828.

Now plug in the values we know into the formula:

\[
600 = P e^{0.04 \times 7}
\]

Now calculate \( 0.04 \times 7 \):

\[
0.04 \times 7 = 0.28
\]

Next, calculate \( e^{0.28} \):

\[
e^{0.28} \approx 1.32313 \quad (\text{using a calculator})
\]

Now replace this value in the equation:

\[
600 = P \cdot 1.32313
\]

To find \( P \), divide both sides by \( 1.32313 \):

\[
P = \frac{600}{1.32313} \approx 453.57
\]

Therefore, the initial amount that Troy earned doing odd jobs is approximately:

\[
\boxed{453.57}
\]

So, Troy earned about $453.57.