Final answer:
The question involves calculating the annual revenue needed for a system with a one-time installation cost and decreasing quarterly operating costs over a 10-year period, considering a 12% compound interest rate. It requires the application of financial mathematics concepts such as geometric series, future value, and present value of annuities.
Step-by-step explanation:
The question relates to financial mathematics, specifically to finding the annual revenue for a system with an initial installation cost and variable quarterly operating costs, taking into account a specific interest rate with compound interest. Considering a 10-year period with a 12% per year interest rate compounded quarterly, the operation costs start at $200,000 for the first quarter of year 1 and decrease by $2000 every subsequent quarter. To calculate the annual revenue, we would use formulas related to geometric series for the operating costs and incorporate the effects of compounding interest for revenue growth.
An initial step would be to calculate the total operating costs over the 10-year period, factoring in the decreasing operating costs each quarter. Then, to find the annual revenue needed to cover these costs plus the installation cost at year 0, we would need to consider the future value of these costs at the interest rate given, since this rate influences the amount of money that needs to be earned to cover the costs when they are incurred in the future.
Finally, we apply financial formulas such as the present value and future value of annuities, to determine how much revenue must be generated annually to result in a future value sufficient to cover the total costs over the 10-year period, given the compounded interest rate.