A municipal wastewater pumping station incurs a maintenance cost of USD 8{,}000 at the end of year 1. The cost increases by USD 1{,}200 in each following year through the end of year 8. At an interest rate of 7% per year, the present worth of the 8-year maintenance stream is most nearly:
- (A)USD 47{,}800
- (B)USD 54{,}900
- (C)USD 65{,}400
- (D)USD 70{,}300
Show worked solution
Answer: (D)
Base series and gradient together discount to a present worth of about USD 70{,}300.
Costs that grow by a constant amount each year separate into a uniform base series of USD 8{,}000 and an arithmetic gradient of USD 1{,}200 whose first increment falls at the end of year 2.
At 7% over eight years the two factors take the values below, with .
Gradient growth supplies almost a third of the obligation, so a reserve sized on the first-year cost alone falls badly short.
FE Reference Handbook — Engineering Economics: Uniform Series Present Worth Factor and Uniform Gradient Present Worth Factor
Why the other choices appear
- (A)Discounting only the uniform USD 8,000 base, 8,000(5.9713) = 47,770, discards the gradient entirely.
- (B)Treating the whole stream as a uniform USD 9,200 (the year-2 amount) gives 9,200(5.9713) = 54,936.
- (C)Using (P/G, 7%, 7) = 14.7149, from counting the seven gradient steps as the factor's period count, gives 47,770 + 17,658 = 65,428.