Question

Manny invests $100 in an account that is compounded continuously at an annual interest rate of 2%, according to the formula A=Pe
rt
, 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?
A. \(5,460 B. \)102 C. \(122 D. \)149

Answer 1

To find how much Manny's investment will be worth in 20 years with continuous compounding, we can use the formula:

\[
A = Pe^{rt}
\]

Where:
- \( P = 100 \) (the principal amount)
- \( r = 0.02 \) (the annual interest rate expressed as a decimal)
- \( t = 20 \) (the time in years)
- \( e \) is the base of the natural logarithm, approximately equal to 2.71828.

Now we can substitute the values into the formula:

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

Calculating \( 0.02 \times 20 \):

\[
0.02 \times 20 = 0.4
\]

Now substituting back into the formula:

\[
A = 100 e^{0.4}
\]

Next, we need to calculate \( e^{0.4} \). Using the approximate value:

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

Now, substituting this value back into the equation:

\[
A = 100 \times 1.49182 \approx 149.182
\]

Rounding to the nearest dollar gives us:

\[
A \approx 149
\]

Therefore, the amount accruing after 20 years will be approximately \( \boxed{149} \).