Suppose that $\$ 2000$ is invested at a rate of $3.1 \%$, 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.

Step 1 ：Given that the principal amount (P) is $2000, the annual interest rate (r) is 3.1% or 0.031 in decimal, the number of times that interest is compounded per year (n) is 2 (since it's compounded semiannually), and the time the money is invested for in years (t) is 5 years.

Step 2 ：We can use the formula for compound interest to find the total amount (A) after 5 years. The formula is \(A = P (1 + \frac{r}{n})^{nt}\)

Step 3 ：Substitute the given values into the formula: \(A = 2000 (1 + \frac{0.031}{2})^{2*5}\)

Step 4 ：Calculate the value inside the parentheses: \(1 + \frac{0.031}{2} = 1.0155\)

Step 5 ：Raise the result to the power of 10 (2 times 5 years): \((1.0155)^{10} = 1.1662704646057548\)

Step 6 ：Multiply the result by the principal amount: \(2000 * 1.1662704646057548 = 2332.5409292115096\)

Step 7 ：Round the final result to the nearest cent: \(\boxed{2332.54}\)