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.

To calculate the future value of the investment using compound interest, you can use the formula: \[ A = P \left(1 + \frac{r}{n}\right)^{nt} \] Where: - \( A \) is the amount of money accumulated after n years, including interest. - \( P \) is the principal amount (the initial amount of money). Here, \( P = 20,000 \). - \( r \) is the annual interest rate (decimal). Here, \( r = 0.075 \). - \( n \) is the number of times that interest is compounded per year. For quarterly, \( n = 4 \). - \( t \) is the time the money is invested for, in years. Here, \( t = 7 \). Plugging in the values: \[ A = 20000 \left(1 + \frac{0.075}{4}\right)^{4 \cdot 7} \] \[ A = 20000 \left(1 + 0.01875\right)^{28} \] \[ A = 20000 \left(1.01875\right)^{28} \] \[ A = 20000 \left(1.6507\right) \] \[ A \approx 33014.00 \] In 7 years there will be \( \$ 33,014.00 \) available. The story of compound interest reaches back centuries! It has roots in Babylonian times, where the concept of earning interest on loans began to take shape. Fast forward to the modern era, and now we see this powerful financial principle shaping investments and retirement plans. Just imagine medieval merchants getting excited about compound interest – it’s like the original investment hack that keeps multiplying wealth! In real-world terms, understanding compound interest is essential for anyone looking to build wealth over time. Whether you’re saving for retirement, a house, or even that shiny new machine, having your money work for you is key. It’s like planting a tree today and enjoying the shade (and fruit) for years to come. So, what are you waiting for? Start investing and let compound interest do its magic!