Step 1 :Given in the problem: Future Value (FV) = $11,600, annual interest rate (r) = 4% = 0.04, number of times that interest is compounded per year (n) = 2, and time the money is invested for in years (t) = 10 years.
Step 2 :The formula for the present value of a future amount is: \(PV = \frac{FV}{(1 + \frac{r}{n})^{nt}}\)
Step 3 :Substitute the given values into the formula: \(PV = \frac{$11,600}{(1 + \frac{0.04}{2})^{2*10}}\)
Step 4 :Calculate the rate per period: \(0.04/2 = 0.02\)
Step 5 :Calculate the total number of periods: \(2*10 = 20\)
Step 6 :Calculate the factor \((1 + rate)^{periods}\): \((1 + 0.02)^{20} \approx 1.48595\)
Step 7 :Divide the future value by this factor: \($11,600 / 1.48595 \approx $7803.64\)
Step 8 :\(\boxed{PV \approx $7803.64}\)
Step 9 :The amount of interest earned is the difference between the future value and the present value: \(Interest Earned = FV - PV = $11,600 - $7803.64 = $3796.36\)
Step 10 :\(\boxed{Interest Earned \approx $3796.36}\)