Euler's Number, e Quick Check

Question

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.