Pat 1 of 3 Which investment results in the greatest total amount? Investment A: \( \$ 4,000 \) invested for 6 years compounded semiannually at \( 6 \% \). Investment B: \( \$ 6,000 \) invested for 4 years compounded quarterly at \( 2.7 \% \). Find the total amount of investment \( A \). \( \$ \square \) (Round to the nearest cent as needed.)

To find the total amount for Investment A compounded semiannually, we can use the formula for compound interest: \[ A = P \left(1 + \frac{r}{n}\right)^{nt} \] Where: - \( A \) is the total amount after time \( t \), - \( P \) is the principal amount (the initial investment), - \( r \) is the annual interest rate (in decimal), - \( n \) is the number of times interest is compounded per year, - \( t \) is the number of years the money is invested. For Investment A: - \( P = 4000 \) - \( r = 0.06 \) (6% as a decimal) - \( n = 2 \) (compounded semiannually) - \( t = 6 \) Now, plug in the values: \[ A = 4000 \left(1 + \frac{0.06}{2}\right)^{2 \times 6} \] \[ = 4000 \left(1 + 0.03\right)^{12} \] \[ = 4000 \left(1.03\right)^{12} \] Calculating \( (1.03)^{12} \): \[ (1.03)^{12} \approx 1.42576 \] So, \[ A \approx 4000 \times 1.42576 \] \[ \approx 5703.04 \] The total amount of Investment A is: \[ \$ 5,703.04 \]