# Feasible region ?

Apr 4, 2017

See below.

#### Explanation:

Calling ${p}_{c} = 52.5$, ${p}_{s} = 37.5$
and defining

${n}_{c} =$number of cement bags
${n}_{s} =$number of sand bags

the feasible quantities obey the restrictions

$\left\{\begin{matrix}{n}_{c} \ge 16 \\ {n}_{s} \ge 8 \\ {n}_{c} + {n}_{s} \ge 105 \\ {n}_{c} + {n}_{s} \le 248\end{matrix}\right.$

The weight function is

$W = {n}_{c} {p}_{c} + {n}_{s} {p}_{s}$

Follows a plot showing the feasible region superimposed with a level weight function representation. The level function representation is done drawing the successive lines in the plane ${n}_{c} , {n}_{s}$ associated to a given total weight

The parallel lines are the plot of ${W}_{i} = {n}_{c} {p}_{c} + {n}_{s} {p}_{s}$ with
${W}_{i} = \left\{{w}_{1} , {w}_{2} , {w}_{2} , \cdots ,\right\}$

In the plot can be observed to the bottom right the maximum weight point attained with ${n}_{c} = 248 , {n}_{s} = 8$ and at the bottom left we have the minimum weight point attained at ${n}_{s} = 89 , {n}_{c} = 16$

The maximum weight is then

${W}_{\max} = 248 \times 52.5 + 8 \times 37.5$

and the minimum

${W}_{\min} = 16 \times 52.5 + 89 \times 37.5$