Question

Tony earned some money doing odd jobs last summer and put it in a savings account that earns 3% interest compounded continuously. After 1 year, there is $100.00 in the account. How much did Tony earn doing odd jobs?
Round your answer to the nearest cent.
$

Answer 1

To find out how much Tony earned doing odd jobs, 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 amount of money),
- \(r\) is the annual interest rate (in decimal form),
- \(t\) is the time in years,
- \(e\) is the base of the natural logarithm (approximately equal to 2.71828).

In this case:
- \(A = 100\),
- \(r = 0.03\) (which is 3% expressed as a decimal),
- \(t = 1\).

We need to solve for \(P\). Plugging the values into the formula, we have:

\[
100 = Pe^{0.03 \times 1}
\]

This simplifies to:

\[
100 = Pe^{0.03}
\]

Now, we need to calculate \(e^{0.03}\):

\[
e^{0.03} \approx 1.03045453395
\]

Now, we can rewrite the equation as:

\[
100 = P \cdot 1.03045453395
\]

To find \(P\), we divide both sides by \(1.03045453395\):

\[
P = \frac{100}{1.03045453395} \approx 97.087
\]

Rounding to the nearest cent, we find:

\[
P \approx 97.09
\]

Therefore, Tony earned approximately **$97.09** doing odd jobs.