Question

Euler's Number, e Quick Check
3 of 53 of 5 Items
Question
Isabella invests $10,000 in an account that is compounded continuously at an annual interest rate of 3%, according to the formula A=Pert
, where A
is the amount accrued, P
is the principal, r is the rate of interest, and t is the time, in years. Apply the formula to find out how much Isabella’s investment will be worth in 15 years, to the nearest dollar.(1 point)
Responses

$13,499
$13,499

$900,171
$900,171

$10,304
$10,304

$15,683

Answer 1

To calculate the future value \( A \) of an investment compounded continuously, you can use the formula:

\[
A = Pe^{rt}
\]

Where:
- \( P \) is the principal (initial investment),
- \( r \) is the annual interest rate (as a decimal),
- \( t \) is the time in years, and
- \( e \) is Euler's number, approximately equal to 2.71828.

Given:
- \( P = 10,000 \)
- \( r = 0.03 \) (3% as a decimal)
- \( t = 15 \)

Now, substitute the values into the formula:

\[
A = 10,000 \times e^{0.03 \times 15}
\]

Calculate \( rt \):

\[
rt = 0.03 \times 15 = 0.45
\]

Now substitute this value back into the formula:

\[
A = 10,000 \times e^{0.45}
\]

Next, calculate \( e^{0.45} \):

Using a calculator:

\[
e^{0.45} \approx 1.5708
\]

Now multiply by the principal:

\[
A \approx 10,000 \times 1.5708 \approx 15,708
\]

Rounding to the nearest dollar, Isabella’s investment will be worth approximately **$15,708**.

Based on the options provided, the closest response would be **$15,683**.