Using technology, determine the present value given that you make quarterly payments of \( \$ 1,635 \) at \( 4.2 \% \) compouded quarterly over a 3 year period. Round your answer to the nearest cent. \( \begin{array}{ll}\text { a. } \$ 15,168.06 & \text { c. } \$ 19,180.84 \\ \text { b. } \$ 19,188.94 & \text { d. } \$ 18,344.05\end{array} \)

To determine the present value of the quarterly payments, we can use the formula for the present value of an annuity: \[ PV = P \times \frac{1 - (1 + r)^{-n}}{r} \] where: - \( PV \) is the present value, - \( P \) is the payment amount per period (\$1,635 in this case), - \( r \) is the interest rate per period (4.2% compounded quarterly, so \( r = 0.042/4 = 0.0105 \)), - \( n \) is the total number of periods (3 years * 4 quarters per year = 12 quarters). Substitute the values into the formula and calculate the present value. Calculate the value by following steps: - step0: Calculate: \(\frac{1635\left(1-\left(1+0.0105\right)^{-12}\right)}{0.0105}\) - step1: Add the numbers: \(\frac{1635\left(1-1.0105^{-12}\right)}{0.0105}\) - step2: Convert the expressions: \(\frac{1635\left(1-\left(\frac{2021}{2000}\right)^{-12}\right)}{0.0105}\) - step3: Subtract the terms: \(\frac{1635\times \frac{2021^{12}-2000^{12}}{2021^{12}}}{0.0105}\) - step4: Multiply the numbers: \(\frac{\frac{1635\times 2021^{12}-1635\times 2000^{12}}{2021^{12}}}{0.0105}\) - step5: Convert the expressions: \(\frac{\frac{1635\times 2021^{12}-1635\times 2000^{12}}{2021^{12}}}{\frac{21}{2000}}\) - step6: Multiply by the reciprocal: \(\frac{1635\times 2021^{12}-1635\times 2000^{12}}{2021^{12}}\times \frac{2000}{21}\) - step7: Rewrite the expression: \(\frac{3\left(545\times 2021^{12}-545\times 2000^{12}\right)}{2021^{12}}\times \frac{2000}{21}\) - step8: Reduce the numbers: \(\frac{545\times 2021^{12}-545\times 2000^{12}}{2021^{12}}\times \frac{2000}{7}\) - step9: Multiply the fractions: \(\frac{\left(545\times 2021^{12}-545\times 2000^{12}\right)\times 2000}{2021^{12}\times 7}\) - step10: Multiply: \(\frac{1090000\times 2021^{12}-1090000\times 2000^{12}}{2021^{12}\times 7}\) The present value of the quarterly payments is approximately \$18,344.05. Therefore, the correct answer is d. \$18,344.05.