The initial cost of a thermal energy system is $650,000. The uniform annual income derived from the system, before maintenance costs, is $215,000. There are no maintenance costs in the first year. However, in the second year of operation, maintenance costs are $12,000. The maintenance costs increase by $12,000 in each subsequent year. The expected rate of return from this system is 15%. The goal of this problem is to determine the life of the system that maximizes the uniform annual profit.

a. Formulate the objective function for this problem. There should be one design variable; the life of the thermal energy system, n.

b. What is the break-even point for this scenario (i.e., at what time is the uniform annual profit zero?)

To solve the unconstrained problem requires the derivative of the objective function with respect to n. However, this derivative can be quite complex to determine since n is involved in economic interest factors. As an alternative, your instructor may assign Part (c), Part (d), or both:

c. Plot the uniform annual profit vs. n and estimate the value of n that maximizes the uniform annual profit. It will be difficult to select a thermal energy system that has an expected life exactly equal to the optimum value. Is it better for the thermal energy system to have a longer life or shorter life compared to the optimum value? Explain your reasoning.

d. Use optimization software to determine the value of n.