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
$\$$
\(\boxed{A = \$967.50}\)
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 :\(\boxed{A = \$967.50}\)