Find the future value and interest earned if $\$ 8704.56$ is invested for 8 years at $4 \%$ compounded (a) semiannually and (b) continuously.

Step 1 ：Given that the principal amount (P) is $8704.56, the annual interest rate (r) is 0.04, and the time (t) is 8 years.

Step 2 ：For part (a), the interest is compounded semiannually, so n = 2. We use the formula for compound interest: \(A = P(1 + \frac{r}{n})^{nt}\)

Step 3 ：Substitute the given values into the formula: \(A = 8704.56(1 + \frac{0.04}{2})^{2*8}\)

Step 4 ：Calculate the future value (A) to get approximately $11949.50

Step 5 ：The interest earned is the future value minus the principal amount: \(11949.50 - 8704.56 = 3244.94\)

Step 6 ：For part (b), the interest is compounded continuously. We use the formula for continuous compound interest: \(A = Pe^{rt}\)

Step 7 ：Substitute the given values into the formula: \(A = 8704.56e^{0.04*8}\)

Step 8 ：Calculate the future value (A) to get approximately $11987.29

Step 9 ：The interest earned is the future value minus the principal amount: \(11987.29 - 8704.56 = 3282.73\)

Step 10 ：\(\boxed{\text{Final Answer: The future value of the investment when compounded semiannually is approximately $11949.50 and the interest earned is approximately $3244.94. The future value of the investment when compounded continuously is approximately $11987.29 and the interest earned is approximately $3282.73.}}\)