Question

Deb is saving up money to buy a car. Deb puts $10,000.00 into an account which earns 2% interest, compounded continuously. How much will she have in the account after 10 years?
Round your answer to the nearest cent.
$

Answer 1

To calculate the future value of an investment with continuously compounded interest, we can use the formula:

\[
A = Pe^{rt}
\]

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 (decimal),
- \(t\) is the time the money is invested for in years,
- \(e\) is Euler's number (approximately equal to 2.71828).

In this case:
- \(P = 10,000\),
- \(r = 0.02\) (2% converted to a decimal),
- \(t = 10\).

Now we can substitute the values into the formula:

\[
A = 10000 \times e^{0.02 \times 10}
\]

Calculating the exponent:

\[
0.02 \times 10 = 0.2
\]

Now, calculate \(e^{0.2}\):

Using \(e^{0.2} \approx 1.2214\) (You can use a scientific calculator for this part.)

Now substitute back into the formula:

\[
A = 10000 \times 1.2214 \approx 12214
\]

Finally, rounding to the nearest cent:

\[
A \approx 12214.00
\]

So, Deb will have approximately **$12,214.00** in the account after 10 years.