$2$ Water Supply Design

This simple problem could be easily solved using only common sense. Solve the

problem by common sense. Then solve the problem using Lagrange multipliers. It should

give the same results.

A small city gets its water supply from three sources: two wells and a small upland

reservoir. The well supplies require pumping, and so are more expensive. Reservoir water

flows directly to the town by gravity. The town needs a total of $20$ mgd $($million gallons per

day$)$ of water. The cost of water is zero if obtained from the reservoir, $$\frac{55}{?}$ million gallons if

from Well$-$field $1,$ and $$\frac{30}{?}$ million gallons if from Well$-$field $2.$ The reservoir has a capacity

of $8$ mgd ; Well$-$field $1$ has a capacity of $25$ mgd $,$ and Well$-$field $2$ has a capacity of $10$ mgd $.$

$($Hint: use the Lagrange multipliers for the last three questions.$)$

a$)$ Find the minimum total cost combination of flows from each source.

b$)$ How much, annually, should the city be willing to pay to expand the capacity of the

reservoir?

c$)$ How much, annually, should the city be willing to pay to expand the capacity of

Well$-$field $1?$

d$)$ How much, annually, should the city be willing to pay to expand the capacity of

Well$-$field $2?$