\( \$ 1,200 \) are deposited into an account with a \( 6.75 \% \) interest rate, compounded quarterly (4 times per year). Find the accumulated amount after 20 years. Hint: \( A=P\left(1+\frac{r}{k}\right)^{k t} \) Round your answer to the nearest cent (hundredth).

To find the accumulated amount after 20 years, we can use the formula for compound interest: \[ A = P\left(1 + \frac{r}{k}\right)^{kt} \] Where: - \( A \) is the accumulated amount - \( P \) is the principal amount (initial deposit) = $1,200 - \( r \) is the annual interest rate = 6.75% = 0.0675 - \( k \) is the number of times interest is compounded per year = 4 (quarterly) - \( t \) is the time in years = 20 Substitute the given values into the formula: \[ A = 1200\left(1 + \frac{0.0675}{4}\right)^{4 \times 20} \] Now, we can calculate the accumulated amount after 20 years. Calculate the value by following steps: - step0: Calculate: \(1200\left(1+\frac{0.0675}{4}\right)^{4\times 20}\) - step1: Divide the terms: \(1200\left(1+\frac{27}{1600}\right)^{4\times 20}\) - step2: Add the numbers: \(1200\left(\frac{1627}{1600}\right)^{4\times 20}\) - step3: Multiply the numbers: \(1200\left(\frac{1627}{1600}\right)^{80}\) - step4: Simplify: \(1200\times \frac{1627^{80}}{1600^{80}}\) - step5: Rewrite the expression: \(400\times 3\times \frac{1627^{80}}{1600^{80}}\) - step6: Rewrite the expression: \(400\times 3\times \frac{1627^{80}}{400^{80}\times 4^{80}}\) - step7: Reduce the numbers: \(3\times \frac{1627^{80}}{400^{79}\times 4^{80}}\) - step8: Multiply the fractions: \(\frac{3\times 1627^{80}}{400^{79}\times 4^{80}}\) The accumulated amount after 20 years is approximately $4,577.06. Therefore, after 20 years, the accumulated amount in the account will be $4,577.06.