Final answer:It takes approximately 7.29 years for an investment of $5,000 to grow to $7,100 at an annual interest rate of 7.5%, compounded quarterly.Explanation:To find the time required for an investment of $5,000 to grow to $7,100 at an interest rate of 7.5% per year, compounded quarterly, we use the formula for compound interest:A = P(1+r/n)(nt)where:A is the amount of money accumulated after n years, including interest.P is the principal amount (initial investment).r is the annual interest rate (decimal).n is the number of times that interest is compounded per year.t is the time the money is invested for, in years.Here, A = $7,100, P = $5,000, r = 0.075 (7.5%), and n = 4 (since the interest is compounded quarterly). We need to solve for t.First, convert the interest rate into a decimal and divide by the number of compound periods per year:r/n = 0.075/4 = 0.01875Substitute the values into the formula and solve for t:$7,100 = $5,000(1 + 0.01875)(4t)Divide both sides by the principal (P) to isolate the growth factor:1.42 = (1 + 0.01875)(4t)Take the natural logarithm of both sides to remove the exponent:ln(1.42) = 4t*ln(1.01875)Now, calculate t:t = ln(1.42) / [4*ln(1.01875)]Performing this calculation gives us:t ≈ 7.29Therefore, it takes approximately 7.29 years for the investment to grow to $7,100 when compounded quarterly at a 7.5% annual interest rate.Note that this value is rounded to the nearest hundredth....

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