Step 1 :Given that the principal amount (P) is $15,000, the annual interest rate (r) is 7% or 0.07, the number of times that interest is compounded per year (n) is 4 (since interest is compounded quarterly), and the time the money is invested for (t) is 6 years.
Step 2 :We use the formula for compound interest, which is \(A = P(1 + \frac{r}{n})^{nt}\), where A is the amount of money accumulated after n years, including interest.
Step 3 :Substitute the given values into the formula: \(A = 15000(1 + \frac{0.07}{4})^{4*6}\)
Step 4 :Calculate the value of A to get the total amount of money accumulated after 6 years, including interest: \(A = 22746.641795875494\)
Step 5 :To find the interest earned, subtract the principal amount from the total amount: \(interest\_earned = A - P = 22746.641795875494 - 15000 = 7746.641795875494\)
Step 6 :Round the interest earned to the nearest cent to get the final answer: \(\boxed{\$7746.64}\)