"All industrial facilities have a network of piping that carries water or other liquids. According to a U.S. Department of Energy study, 16% of a typical facility's electricity costs are for its pumping systems. Suppose that you are considering a pumping facility having 11,000 ft to carry 550 gpm of water continuously to storage tank. The general equation for estimating annual pumping costs is
Cost($)= [(Friction factor)(Pipe length in ft)(Number of hours)($/kWh)(Flow in gpm)^3]/[1705*(Combined pump and motor efficiency)(Pipe inner diameter in inches)^5]
Where the friction factor is 0.0058. The electricity cost is $0.06/kWh, the pumping system operates for 7,770 hours annually, and the combined pump and motor efficiency is 0.7. The price for 11,000 feet of steel pipe is $11,000 per inch of the inner diameter. Determine the optimal size of the pipe's inner diameter in inches by minimizing the annual equivalent cost. Assume that the pumping operation will last 18 years, the salvage value of the pipe is 8% of the initial investment, and a discount rate of 12%."