Find the compounded amount of an investment of \( \$ 7,000 \) at \( 8 \% \) interest, compounded quarterly, for 9 years. Select one: O. \( \$ 12,665.23 \) o.b. \( \$ 14,279.29 \) O c. \( \$ 10,385.58 \) o d. \( \$ 16,989.00 \)

To find the compounded amount of an investment, 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: - \( P = 7000 \) - \( r = 0.08 \) (8% as a decimal) - \( n = 4 \) (quarterly compounding) - \( t = 9 \) Now, we can substitute these values into the formula step by step. 1. Calculate \( \frac{r}{n} \): \[ \frac{r}{n} = \frac{0.08}{4} = 0.02 \] 2. Calculate \( nt \): \[ nt = 4 \times 9 = 36 \] 3. Substitute these values into the formula: \[ A = 7000 \left(1 + 0.02\right)^{36} \] 4. Calculate \( 1 + 0.02 \): \[ 1 + 0.02 = 1.02 \] 5. Now calculate \( (1.02)^{36} \): \[ (1.02)^{36} \] 6. Finally, multiply by \( 7000 \): \[ A = 7000 \times (1.02)^{36} \] Let's perform the calculations to find the final amount \( A \). Calculate the value by following steps: - step0: Calculate: \(7000\times 1.02^{36}\) - step1: Convert the expressions: \(7000\left(\frac{51}{50}\right)^{36}\) - step2: Simplify: \(7000\times \frac{51^{36}}{50^{36}}\) - step3: Rewrite the expression: \(50\times 140\times \frac{51^{36}}{50^{36}}\) - step4: Reduce the numbers: \(140\times \frac{51^{36}}{50^{35}}\) - step5: Rewrite the expression: \(10\times 14\times \frac{51^{36}}{50^{35}}\) - step6: Rewrite the expression: \(10\times 14\times \frac{51^{36}}{10^{35}\times 5^{35}}\) - step7: Reduce the numbers: \(14\times \frac{51^{36}}{10^{34}\times 5^{35}}\) - step8: Rewrite the expression: \(2\times 7\times \frac{51^{36}}{10^{34}\times 5^{35}}\) - step9: Rewrite the expression: \(2\times 7\times \frac{51^{36}}{2^{34}\times 5^{34}\times 5^{35}}\) - step10: Reduce the numbers: \(7\times \frac{51^{36}}{2^{33}\times 5^{34}\times 5^{35}}\) - step11: Multiply the fractions: \(\frac{7\times 51^{36}}{2^{33}\times 5^{69}}\) After performing the calculations, we find that the compounded amount \( A \) is approximately: \[ A \approx 14279.21 \] Thus, the compounded amount of the investment of \( \$ 7,000 \) at \( 8\% \) interest, compounded quarterly for 9 years, is approximately \( \$ 14,279.29 \). The correct answer is: **b. \( \$ 14,279.29 \)**.