\( \$ 1,200 \) 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 calculate the accumulated amount after 20 years using the formula \( A = P\left(1+\frac{r}{k}\right)^{k t} \), we can plug in the values: - \( P = 1,200 \) (the principal amount), - \( r = 0.0675 \) (the annual interest rate as a decimal), - \( k = 4 \) (the number of times interest is compounded per year), - \( t = 20 \) (the number of years). Now substituting these values into the formula: \[ A = 1200\left(1+\frac{0.0675}{4}\right)^{4 \times 20} \] Calculate \( \frac{0.0675}{4} \): \[ \frac{0.0675}{4} = 0.016875 \] Now add 1: \[ 1 + 0.016875 = 1.016875 \] Now calculate \( 4 \times 20 = 80 \): Now we calculate \( A \): \[ A = 1200 \left(1.016875\right)^{80} \] Calculating \( (1.016875)^{80} \): \[ (1.016875)^{80} \approx 3.735501 \] Now multiply by \( 1,200 \): \[ A \approx 1200 \times 3.735501 \approx 4482.6012 \] Finally, rounding to the nearest cent: \[ A \approx 4482.60 \] So, the accumulated amount after 20 years is **\( \$4482.60 \)**.