A newborn child receives a $\$ 4,000$ gift toward a college education from her grandparents. How much will the $\$ 4,000$ be worth in 18 years if it is invested at $6.2 \%$ compounded quarterly?

Step 1 ：Translate the problem into the formula for future value (FV) of an investment, which is: \(FV = PV * (1 + r/n)^{nt}\), where PV is the present value (the initial amount of money), 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 time the money is invested for in years.

Step 2 ：Substitute the given values into the formula: \(PV = 4000\), \(r = 6.2/100 = 0.062\), \(n = 4\) (quarterly compounding), and \(t = 18\) years.

Step 3 ：Calculate the future value (FV) using the formula: \(FV = 4000 * (1 + 0.062/4)^{4*18}\)

Step 4 ：Simplify the calculation to get the final answer: \(FV = 12106.39357739892\)

Step 5 ：Round the final answer to two decimal places: \(\boxed{12106.39}\)