A certain objective function is the production function. In utilizing groundwater, we will have a certain number of wells to be utilized for irrigation or domestic uses. The cost of production of water $($pumping$)$ usually takes the form:C $=$ a $+$ pQ $+$ kQ$?$Where: a is a constant fixed cost $($maintenance$),$ Q is the discharge $($MGD$),$ p is the cost usually directly proportional to flow rate and k is the constant which depends on $($friction f$,$ topography, pipe size, roughness e$).$ By increasing the number of wells and increasing the supply, we will have the well field, $($N wells$)$ then we may have M$.$ Design criteria $($maximum discharge Q at minimum costC$).$Question: For the above equation suppose you have:a $=13,$ p $=4,$ k $=1,$ C $=$ cost in $$/$day$,$ Q $=$ million gallon$/$day$($MGD$)(2$ Q $5).$Analyze the economy of the production of water. Check three methods:

a$.$ Analytical method

b$.$ Lagrange method

c$.$ Graphical method.