Final answer:
The question involves calculating compound interest for an investment and the depreciation of a welding machine over time, using appropriate mathematical formulas to determine the future value of the investment and the remaining value of the machine.
Step-by-step explanation:
The question is concerning two topics within mathematics: compound interest and depreciation. Sarah's investment scenario involves compound interest where the interest is calculated monthly at a rate of 5%. To calculate the future value of her investment after 6 years, we can use the formula for compound interest, which is A = P(1 + r/n)^(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, and t is the time in years.
The second scenario involves calculating the value of a welding machine after 12 years with a steady depreciation rate of 8% per year. The formula used for depreciation is V = P(1 - r)^t, where V is the value after t years, P is the original principal amount, r is the annual depreciation rate (decimal), and t is the time in years.
This interest, though, is not re-invested and so the principal remains the same and, if interest rates do not change, the same amount of interest is paid again at the end of the next year. Compound interest is when the interest paid each year is not paid out but is added to the principal.