14 15 16 17 18 19 20 21 22 23 24 To pay for new machinery in 7 years, a company owner invests \( \$ 20,000 \) at \( 7.5 \% \) compounded quarterly. How much money will be available in 7 years? Round your answer to the nearest cent. In 7 years there will be \( \$ \) \( \square \) available. \( \square \)

To find the future value of the investment, you can use the formula for compound interest: \[ A = P \left(1 + \frac{r}{n}\right)^{nt} \] where: - \( A \) is the amount of money accumulated after n years, including interest. - \( P \) is the principal amount (the initial amount of money). - \( r \) is the annual interest rate (decimal). - \( n \) is the number of times that interest is compounded per year. - \( t \) is the time the money is invested for in years. Plugging in the values: - \( P = 20000 \) - \( r = 0.075 \) - \( n = 4 \) (quarterly) - \( t = 7 \) Calculating it step by step: \[ A = 20000 \left(1 + \frac{0.075}{4}\right)^{4 \times 7} \] Calculating the interest per compounding period: \[ A = 20000 \left(1 + 0.01875\right)^{28} \] \[ A = 20000 \left(1.01875\right)^{28} \] Now calculating \( (1.01875)^{28} \): \[ (1.01875)^{28} \approx 1.646619 \] Now, plugging it back into the formula: \[ A \approx 20000 \times 1.646619 \approx 32932.39 \] So, in 7 years there will be \( \$ 32,932.39 \) available.