Question

Suppose that $2000 is invested at a rate of 4%, compounded semiannually. Assuming that no withdrawals are made, find the total amount after 9 years.
Do not round any intermediate computations, and round your answer to the nearest cent.

Answer 1

To find the total amount after 9 years with the investment compounded semiannually, we can use the formula for compound interest:

\[
A = P \left(1 + \frac{r}{n}\right)^{nt}
\]

where:
- \(A\) is the amount of money accumulated after n years, including interest.
- \(P\) is the principal amount (the initial amount of money).
- \(r\) is the annual interest rate (decimal).
- \(n\) is the number of times that interest is compounded per unit \(t\).
- \(t\) is the time the money is invested for, in years.

For this problem:
- \(P = 2000\)
- \(r = 0.04\) (which is 4% in decimal)
- \(n = 2\) (since the interest is compounded semiannually)
- \(t = 9\)

Now we can substitute these values into the formula:

\[
A = 2000 \left(1 + \frac{0.04}{2}\right)^{2 \times 9}
\]

First, calculate \(\frac{0.04}{2} = 0.02\). Now we can plug this back into the equation:

\[
A = 2000 \left(1 + 0.02\right)^{18}
\]
\[
A = 2000 \left(1.02\right)^{18}
\]

Next, we calculate \((1.02)^{18}\). We can either calculate it using a calculator or do it step by step. Using a calculator, we find:

\[
(1.02)^{18} \approx 1.396619
\]

Now, substituting this value back into the equation for \(A\):

\[
A = 2000 \times 1.396619 \approx 2793.238
\]

Rounding to the nearest cent, the total amount after 9 years is:

\[
\boxed{2793.24}
\]