Question

Suppose that $2000 is invested at a rate of 4.6%, compounded semiannually. Assuming that no withdrawals are made, find the total amount after 3 years.
Do not round any intermediate computations, and round your answer to the nearest cent.

Answer 1

To calculate the total amount after 3 years for an investment compounded semiannually, we can use the compound interest formula:

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

where:
- \( A \) is the total amount after \( t \) years,
- \( P \) is the principal amount (the initial investment),
- \( r \) is the annual interest rate (as a decimal),
- \( n \) is the number of times the interest is compounded per year, and
- \( t \) is the number of years the money is invested for.

Given:
- \( P = 2000 \),
- \( r = 4.6\% = 0.046 \),
- \( n = 2 \) (because it's compounded semiannually),
- \( t = 3 \).

Substituting these values into the formula:

\[
A = 2000 \left(1 + \frac{0.046}{2}\right)^{2 \times 3}
\]

Calculating the individual components:

1. Calculate \( \frac{0.046}{2} \):

\[
\frac{0.046}{2} = 0.023
\]

2. Now calculate \( 1 + 0.023 \):

\[
1 + 0.023 = 1.023
\]

3. Calculate \( nt \):

\[
nt = 2 \times 3 = 6
\]

Putting this all together, we have:

\[
A = 2000 \left(1.023\right)^{6}
\]

Next, we evaluate \( (1.023)^{6} \):

\[
(1.023)^{6} \approx 1.148882
\]

Now substituting this value back:

\[
A = 2000 \times 1.148882
\]

Calculating this multiplication:

\[
A \approx 2297.764
\]

Finally, rounding to the nearest cent gives us:

\[
A \approx 2297.76
\]

Thus, the total amount after 3 years is

\[
\boxed{2297.76}
\]