Find the time required for an investment of 5000 dollars to grow to 6600 dollars at an interest rate of 7.5 percent per year, compounded quarterly.
Your answer is $t=$ years. You may enter the exact value or round to 2 decimal places.

Step 1 ：We are given that the principal amount (P) is 5000 dollars, the amount of money accumulated after n years (A) is 6600 dollars, the annual interest rate (r) is 7.5 percent or 0.075 in decimal form, and the number of times that interest is compounded per year (n) is 4 (since the interest is compounded quarterly). We need to find the time the money is invested for in years (t).

Step 2 ：The formula for compound interest is \(A = P(1 + r/n)^{nt}\).

Step 3 ：We can rearrange this formula to solve for t, giving us \(t = \frac{\log(A/P)}{n \log(1 + r/n)}\).

Step 4 ：Substituting the given values into this formula, we get \(t = \frac{\log(6600/5000)}{4 \log(1 + 0.075/4)}\).

Step 5 ：Calculating this gives us \(t = 3.7363530099575764\).

Step 6 ：Rounding this to two decimal places gives us \(t = 3.74\).

Step 7 ：Final Answer: The time required for an investment of 5000 dollars to grow to 6600 dollars at an interest rate of 7.5 percent per year, compounded quarterly is approximately \(\boxed{3.74}\) years.