As the prize in a contest, you are offered $\$ 16,000$ now or $\$ 18,000$ in 5 years. If the money can be invested at $6 \%$ compounded annually, which prize will be worth more in 5 years? Choose the option which is worth more after 5 years accepting a $\$ 16,000$ prize now and investing it at $6 \%$ compounded annually accepting a $\$ 18,000$ prize in 5 years

Step 1 ：We are given two options: accepting a $16,000 prize now and investing it at 6% compounded annually, or accepting an $18,000 prize in 5 years. We need to determine which option will be worth more after 5 years.

Step 2 ：First, we calculate the future value of the $16,000 if it is invested at a 6% annual interest rate for 5 years. The future value of an investment is calculated using the formula: \(FV = PV \times (1 + r/n)^{nt}\), where: \(FV\) is the future value, \(PV\) is the present value, \(r\) is the annual interest rate (in decimal form), \(n\) is the number of times that interest is compounded per year, and \(t\) is the number of years the money is invested for.

Step 3 ：In this case, the present value (PV) is $16,000, the annual interest rate (r) is 6% or 0.06, the number of times that interest is compounded per year (n) is 1 (since it's compounded annually), and the number of years the money is invested for (t) is 5.

Step 4 ：Substituting these values into the formula, we get: \(FV = 16000 \times (1 + 0.06/1)^{1 \times 5}\)

Step 5 ：Solving this equation, we find that the future value of the $16,000 investment after 5 years, when compounded annually at a rate of 6%, is approximately $21,411.61.

Step 6 ：Now, we compare this value with the $18,000 that would be received in 5 years. Since $21,411.61 is greater than $18,000, the $16,000 investment would be worth more in 5 years.

Step 7 ：Final Answer: The option of accepting a $16,000 prize now and investing it at 6% compounded annually is worth more after 5 years. The future value of this option is approximately \(\boxed{\$21,411.61}\).