Problem $4$

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:

$C=a+pQ+k{Q}^2$

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 cost C$).$

Question: For the above equation suppose you have:

$)Q(5$

Analyze the economy of the production of water. Check three methods:

a$.$ Analytical method

b$.$ Lagrange method

c$.$ Graphical method.