Suppose that $2000 is invested at a rate of 4.7%, compounded semiannually. Assuming that no withdrawals are made, find the total amount after 5 years.Do not round any intermediate computations, and round your answer to the nearest cent.$

To find the total amount after 5 years for an investment of $2000 at a rate of 4.7%, compounded semiannually, we can use the compound interest formula:

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

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 year.
• $t$ is the time the money is invested for in years.

Given:

• $P = 2000$
• $r = 4.7\% = 0.047$
• $n = 2$ (since the interest is compounded semiannually)
• $t = 5$

Substitute these values into the formula:

$A = 2000 \left(1 + \frac{0.047}{2}\right)^{2 \cdot 5}$

First, calculate the semiannual interest rate:

$\frac{0.047}{2} = 0.0235$

Next, calculate the exponent:

$2 \cdot 5 = 10$

Now, compute the expression inside the parentheses:

$1 + 0.0235 = 1.0235$

Raise this to the power of 10:

$1.0235^{10} \approx 1.268241794$

Finally, multiply by the principal amount:

$A = 2000 \times 1.268241794 \approx 2536.48$

So, the total amount after 5 years, rounded to the nearest cent, is:

$\boxed{2536.48}$