\"A building owner is considering installing solar roof panels on a building. Th

Question

"A building owner is considering installing solar roof panels on a building. The cost of installing solar panels is $21,000 $18.30*S where S is the square footage of the solar panels. There is an annual maintenance cost of $3.00 per square foot of installed solar panel. The solar panels will last 22 years, and there is no salvage value or disposal cost. The electricity usage for a single hour for the building with operating solar solar panels is E = 1000/S where E is the electricity usage in kWh (kilowatt-hour) and S is the square footage of the solar panels. The cost of electricity is $0.20 per kWh, and the solar panels will operate for 15 hours per day 365 days per year. The building owner's MARR is 8.4%. Assuming the building owner wants to minimize the annual equivalent cost, how many square feet of solar panels should be installed? HINT: You should convert the cost of installing the solar panel to an annual cost."

Explanation / Answer

Cost of installing panel = 21,000 + 18.30 S

Distribute this installing cost over 22 years

PV = (PMT/0.084)*(1 – 1/1.084^22) = 9.886040 PMT = A*PMT

PMT = (21,000 + 18.30 S)/A where A = 9.886040

Annual maintenance cost = 3 S

Annual electricity cost = (1000/S)*0.20*15*365 = 1,095,000/S

Total annual cost = C = 21000/A + 18.30S/A + 3S + 1,095,000/S

Differentiate with respect to S

dC/dS = 18.30/A + 3 - 1,095,000/S^2

Again differentiate with respect to S

d2C/dS2 = 2*1,095,000/S^3

C is minimum when d2C/dS2 > 0; hence we need S>0

For optimal value, dC/dS = 18.30/A + 3 - 1,095,000/S^2 = 0

1,095,000/S^2 = 18.30/A + 3

S^2 = 1,095,000/(18.30/A + 3) = 225,722.229741

S = ±475.10

For C to be minimal, S must be 475.10 square-footage

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