Olivia would like to buy some new furniture for her home. She decides to buy the furniture on credit with \( 9.5 \% \) interest compounded quarterly. If she spent \( \$ 7.400 \), how much total will she have paid after 8 years? \( \$ 15.415 .94 \) \( \$ 16.349 .72 \) \( \$ 15,927.56 \) \( \$ 15,683.28 \)

To find out how much Olivia will have paid after 8 years for her furniture purchase on credit with interest compounded quarterly, we can use the formula for compound interest: \[ A = P \left(1 + \frac{r}{n}\right)^{nt} \] Where: - \( A \) is the amount of money accumulated after n years, including interest. - \( P \) is the principal amount (the initial amount of money). - \( r \) is the annual interest rate (decimal). - \( n \) is the number of times that interest is compounded per year. - \( t \) is the number of years the money is invested or borrowed. Given: - \( P = 7400 \) - \( r = 9.5\% = 0.095 \) - \( n = 4 \) (since the interest is compounded quarterly) - \( t = 8 \) Now, we can substitute these values into the formula to calculate \( A \): \[ A = 7400 \left(1 + \frac{0.095}{4}\right)^{4 \times 8} \] Let's calculate this step by step. 1. Calculate \( \frac{r}{n} \): \[ \frac{0.095}{4} \] 2. Calculate \( nt \): \[ 4 \times 8 \] 3. Substitute these values into the formula and calculate \( A \). Now, I will perform these calculations. Calculate the value by following steps: - step0: Calculate: \(4\times 8\) - step1: Multiply the numbers: \(32\) Calculate or simplify the expression \( 0.095/4 \). Calculate the value by following steps: - step0: Calculate: \(\frac{0.095}{4}\) - step1: Convert the expressions: \(\frac{\frac{19}{200}}{4}\) - step2: Multiply by the reciprocal: \(\frac{19}{200}\times \frac{1}{4}\) - step3: Multiply the fractions: \(\frac{19}{200\times 4}\) - step4: Multiply: \(\frac{19}{800}\) Calculate or simplify the expression \( 7400*(1+0.02375)^(32) \). Calculate the value by following steps: - step0: Calculate: \(7400\left(1+0.02375\right)^{32}\) - step1: Add the numbers: \(7400\times 1.02375^{32}\) - step2: Convert the expressions: \(7400\left(\frac{819}{800}\right)^{32}\) - step3: Simplify: \(7400\times \frac{819^{32}}{800^{32}}\) - step4: Rewrite the expression: \(200\times 37\times \frac{819^{32}}{800^{32}}\) - step5: Rewrite the expression: \(200\times 37\times \frac{819^{32}}{200^{32}\times 4^{32}}\) - step6: Reduce the numbers: \(37\times \frac{819^{32}}{200^{31}\times 4^{32}}\) - step7: Multiply the fractions: \(\frac{37\times 819^{32}}{200^{31}\times 4^{32}}\) Now that we have completed the calculations, we can summarize the results: 1. The quarterly interest rate \( \frac{r}{n} \) is approximately \( 0.02375 \). 2. The total number of compounding periods \( nt \) is \( 32 \). Now, substituting these values into the compound interest formula: \[ A = 7400 \left(1 + 0.02375\right)^{32} \] Calculating this gives us: \[ A \approx 7400 \times (1.02375)^{32} \approx 15683.28 \] Thus, the total amount Olivia will have paid after 8 years is approximately \( \$ 15,683.28 \). The correct answer is \( \$ 15,683.28 \).