If $\$ 38600$ is invested at an interest rate of 7 percent per year, find the value of the investment at the end of 5 years for the following compounding methods. Round answers to the nearest cent.
Compounded
(a) annually:
(b) semiannually:
(c) monthly:
(d) daily:
(e) continuously: \$

Step 1 ：Given that the principal amount (P) is \$38600, the annual interest rate (r) is 7% or 0.07 in decimal form, and the time (t) the money is invested for is 5 years.

Step 2 ：We are asked to find the value of the investment at the end of 5 years for different compounding methods. The formula for compound interest varies depending on the compounding method.

Step 3 ：For annually compounding (a), the formula is \(A = P(1 + \frac{r}{n})^{nt}\), where n is the number of times that interest is compounded per year. In this case, n is 1 as interest is compounded once a year. Substituting the given values into the formula, we get \(A = 38600(1 + \frac{0.07}{1})^{1*5}\), which simplifies to \(A = 54138.49680502001\). Rounding to the nearest cent, we get \(\boxed{54138.50}\).

Step 4 ：For semiannually compounding (b), n is 2 as interest is compounded twice a year. Substituting the given values into the formula, we get \(A = 38600(1 + \frac{0.07}{2})^{2*5}\), which simplifies to \(A = 54449.11215997527\). Rounding to the nearest cent, we get \(\boxed{54449.11}\).

Step 5 ：For monthly compounding (c), n is 12 as interest is compounded twelve times a year. Substituting the given values into the formula, we get \(A = 38600(1 + \frac{0.07}{12})^{12*5}\), which simplifies to \(A = 54720.33502110002\). Rounding to the nearest cent, we get \(\boxed{54720.34}\).

Step 6 ：For daily compounding (d), n is 365 as interest is compounded 365 times a year. Substituting the given values into the formula, we get \(A = 38600(1 + \frac{0.07}{365})^{365*5}\), which simplifies to \(A = 54774.169268703285\). Rounding to the nearest cent, we get \(\boxed{54774.17}\).

Step 7 ：For continuously compounding (e), the formula is \(A = Pe^{rt}\), where e is the base of the natural logarithm (approximately equal to 2.71828). Substituting the given values into the formula, we get \(A = 38600e^{0.07*5}\), which simplifies to \(A = 54776.00737569973\). Rounding to the nearest cent, we get \(\boxed{54776.01}\).

Step 8 ：Final Answer: The value of the investment at the end of 5 years is (a) annually: \(\boxed{54138.50}\), (b) semiannually: \(\boxed{54449.11}\), (c) monthly: \(\boxed{54720.34}\), (d) daily: \(\boxed{54774.17}\), (e) continuously: \(\boxed{54776.01}\).