Question

Use the compound interest formulas A=P (1+r/n)^nt and A =Pe^rt to solve the problem given. Round answers to the nearest cent.

Find the accumulated value of an investment of $20,000 for 6 years at an interest rate of 7% if the money is a.
compounded semiannually; b. compounded quarterly; c. compounded monthly; d. compounded continuously.

a. What is the accumulated value if the money is compounded semiannually?

$ blank (Round your answer to the nearest cent. Do not include the $ symbol in your answer.)

Answer 1

To calculate the accumulated value of an investment using the compound interest formulas, we will use:

1. **For semiannual compounding**:
\[
A = P \left(1 + \frac{r}{n}\right)^{nt}
\]
where:
- \( P = 20000 \) (the principal)
- \( r = 0.07 \) (the annual interest rate in decimal)
- \( n = 2 \) (the number of times interest is compounded per year)
- \( t = 6 \) (the number of years)

Substituting in the values for semiannual compounding:

\[
A = 20000 \left(1 + \frac{0.07}{2}\right)^{2 \times 6}
\]
\[
A = 20000 \left(1 + 0.035\right)^{12}
\]
\[
A = 20000 \left(1.035\right)^{12}
\]

Now we calculate \( (1.035)^{12} \):

\[
(1.035)^{12} \approx 1.491825
\]

Now we multiply by the principal:

\[
A \approx 20000 \times 1.491825 \approx 29836.50
\]

So the accumulated value if the money is compounded semiannually is:

\[
\boxed{29836.50}
\]