Question

Manny invests $100 in an account that is compounded continuously at an annual interest rate of 2%, 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. In 20 years, how much will Manny’s investment be worth to the nearest dollar?(1 point)
Responses

$102
$102

$5,460
$5,460

$122
$122

$149

Answer 1

To find out how much Manny's investment will be worth in 20 years with continuous compounding at an annual interest rate of 2%, we can use the formula:

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

Where:
- \( A \) is the amount accrued,
- \( P \) is the principal (initial investment),
- \( r \) is the rate of interest (in decimal),
- \( t \) is the time in years,
- \( e \) is the base of the natural logarithm, approximately equal to 2.71828.

Plugging in the values:
- \( P = 100 \)
- \( r = 0.02 \) (2% as a decimal)
- \( t = 20 \)

Now we can calculate \( A \):

\[
A = 100 \cdot e^{0.02 \cdot 20}
\]
\[
A = 100 \cdot e^{0.4}
\]

Now we need to calculate \( e^{0.4} \):

Using a calculator, we find:

\[
e^{0.4} \approx 1.49182
\]

Now substituting that back into the equation:

\[
A \approx 100 \cdot 1.49182 \approx 149.18
\]

Rounding to the nearest dollar:

\[
A \approx 149
\]

Thus, Manny’s investment will be worth approximately **$149** in 20 years.