How much money will there be in an account at the end of 6 years if \( \$ 14000 \) is deposited at \( 3 \% \) interest compounded quarterly? (Assume no withdrawals are made.) \( \begin{array}{l}\text { The amount after } 6 \text { years will be } \$ \square \text {. } \\ \text { (Round to the nearest cent as needed.) }\end{array} \)

To calculate the amount of money in the account at the end of 6 years with a deposit of $14000 at an interest rate of 3% 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 in the account after \( t \) years - \( P \) is the principal amount (initial deposit) - \( r \) is the annual interest rate (in decimal form) - \( n \) is the number of times the interest is compounded per year - \( t \) is the number of years Given: - \( P = \$14000 \) - \( r = 3\% = 0.03 \) - \( n = 4 \) (compounded quarterly) - \( t = 6 \) years Substitute these values into the formula and calculate the amount of money in the account after 6 years. Calculate the value by following steps: - step0: Calculate: \(14000\left(1+\frac{0.03}{4}\right)^{4\times 6}\) - step1: Divide the terms: \(14000\left(1+\frac{3}{400}\right)^{4\times 6}\) - step2: Add the numbers: \(14000\left(\frac{403}{400}\right)^{4\times 6}\) - step3: Multiply the numbers: \(14000\left(\frac{403}{400}\right)^{24}\) - step4: Simplify: \(14000\times \frac{403^{24}}{400^{24}}\) - step5: Rewrite the expression: \(400\times 35\times \frac{403^{24}}{400^{24}}\) - step6: Reduce the numbers: \(35\times \frac{403^{24}}{400^{23}}\) - step7: Rewrite the expression: \(5\times 7\times \frac{403^{24}}{400^{23}}\) - step8: Rewrite the expression: \(5\times 7\times \frac{403^{24}}{25^{23}\times 16^{23}}\) - step9: Rewrite the expression: \(5\times 7\times \frac{403^{24}}{5^{46}\times 16^{23}}\) - step10: Reduce the numbers: \(7\times \frac{403^{24}}{5^{45}\times 16^{23}}\) - step11: Multiply the fractions: \(\frac{7\times 403^{24}}{5^{45}\times 16^{23}}\) The amount of money in the account at the end of 6 years will be approximately $16749.79.