A pipe network consists of a vertical pipe which draws water from a well that is open to the atmosphere, a pump, and a $500-$ft long horizontal pipe which discharges to an open$-$topped tank through a submerged exit. Both pipe sections are smooth and have the same diameter, which can be specified in $$ increments from $1$ to $3.$ The pipe costs $$0\mathrm{.}15($LD$^2),$ where L is the pipe length in ft and D is the diameter in inches. The pump is located at ground level between the vertical and horizontal sections of pipe. The pump curve is h$\_$p $=\backslash$beta $(5-0\mathrm{.}001$Q$^2)$ where h$\_$p is the pump head in feet, Q is the flowrate in gpm$,$ and $\backslash$beta is a dimensionless parameter that specifies the size of pump$/$motor combination chosen $($for simplicity, $\backslash$beta can have any value from $1$ to $10).$ The cost of the pump$/$motor is $$100\backslash$beta $.$ If the flowrate is $35$ gpm and the well depth is $15$ ft$,$ determine the pipe diameter D and pump$/$motor size $\backslash$beta that minimizes the capital cost of the system. $($The depth of the well is measured from the surface of the storage tank to the surface of the well.$)$ You will have to solve this problem numerically by calculating the total capital cost over the range of allowable diameters.