If $\$ 500$ is invested at an interest rate of $5.5 \%$ per year, find the amount of the investment at the end of 12 years for the following compounding methods. (Round your answers to the nearest cent.) (a) Annually $\$$ (b) Semiannually $\$$ (c) Quarterly $\$$ (d) Continuously $\$$

Step 1 ：\(A = P(1 + r/n)^{nt}\)

Step 2 ：For annually, \(n = 1\). So, the formula becomes \(A = P(1 + r)^t\)

Step 3 ：Substituting the given values, we get \(A = 500(1 + 0.055/1)^{1*12}\)

Step 4 ：Solving the equation, we get \(A = 500(1.055)^{12}\)

Step 5 ：Further simplifying, we get \(A = 500 * 1.8983\)

Step 6 ：\(\boxed{A = \$949.15}\)

Step 7 ：For semiannually, \(n = 2\). So, the formula becomes \(A = P(1 + r/2)^{2t}\)

Step 8 ：Substituting the given values, we get \(A = 500(1 + 0.055/2)^{2*12}\)

Step 9 ：Solving the equation, we get \(A = 500(1.0275)^{24}\)

Step 10 ：Further simplifying, we get \(A = 500 * 1.9051\)

Step 11 ：\(\boxed{A = \$951.55}\)

Step 12 ：For quarterly, \(n = 4\). So, the formula becomes \(A = P(1 + r/4)^{4t}\)

Step 13 ：Substituting the given values, we get \(A = 500(1 + 0.055/4)^{4*12}\)

Step 14 ：Solving the equation, we get \(A = 500(1.01375)^{48}\)

Step 15 ：Further simplifying, we get \(A = 500 * 1.9074\)

Step 16 ：\(\boxed{A = \$953.70}\)

Step 17 ：The formula for continuously compounded interest is \(A = Pe^{rt}\)

Step 18 ：Substituting the given values, we get \(A = 500e^{0.055*12}\)

Step 19 ：Solving the equation, we get \(A = 500 * e^{0.66}\)

Step 20 ：Further simplifying, we get \(A = 500 * 1.9350\)

Step 21 ：\(\boxed{A = \$967.50}\)