\( 1 \leftarrow \) Find the value of each of these investments at the end of the period specified below. (a) \( \$ 3500 \) invested at \( 5 \frac{2}{5} \% \) compounded annually for five years (b) \( \$ 6000 \) invested at \( 7 \% \) compounded semiannually for ten years

To find the future value of investments compounded annually and semiannually, you can use the formula: \[ F = P \left(1 + \frac{r}{n}\right)^{nt} \] where: - \( F \) is the future value of the investment, - \( P \) is the principal (initial amount), - \( r \) is the annual interest rate (as a decimal), - \( n \) is the number of times the interest is compounded per year, - \( t \) is the number of years the money is invested. ### (a) Future value of $3500 invested at \( 5\frac{2}{5}\% \) or \( 5.4\% \) 1. Convert the interest rate to a decimal: \( 5\frac{2}{5}\% = \frac{27}{5} = 5.4\% = 0.054 \). 2. Principal \( P = 3500 \), \( r = 0.054 \), \( n = 1 \) (compounded annually), \( t = 5 \). Now plug in the values: \[ F = 3500 \left(1 + \frac{0.054}{1}\right)^{1 \times 5} \] \[ F = 3500 \left(1 + 0.054\right)^{5} = 3500 \times (1.054)^{5} \] Calculating \( (1.054)^{5} \approx 1.3005 \). Now, calculate: \[ F \approx 3500 \times 1.3005 \approx 4551.75 \] So, the future value of the investment after 5 years is approximately **$4551.75**. ### (b) Future value of $6000 invested at \( 7\% \) compounded semiannually 1. Convert the interest rate to a decimal: \( 7\% = 0.07 \). 2. Principal \( P = 6000 \), \( r = 0.07 \), \( n = 2 \) (compounded semiannually), \( t = 10 \). Now plug in the values: \[ F = 6000 \left(1 + \frac{0.07}{2}\right)^{2 \times 10} \] \[ F = 6000 \left(1 + 0.035\right)^{20} = 6000 \times (1.035)^{20} \] Calculating \( (1.035)^{20} \approx 1.8061 \). Now, calculate: \[ F \approx 6000 \times 1.8061 \approx 10836.60 \] So, the future value of the investment after 10 years is approximately **$10,836.60**.