Question

The corner points of the feasible region determined by the system of linear constraints are (0,10), (5,5), (15,15), (0,20). Let$z = px + qy$ where $p,q > 0$ condition on $p$ and $q$ so that the maximum of $z$ occurs at both the points (15,15) and (0,20) is ____________
A. $q = 2p$
B. $p = 2q$
C. $p = q$
D. $q = 3p$

Answer 1

Here, we would be using the fact that the maximum value attained by $z$at any corner point of the feasible region is equal.

Complete step-by-step answer:
Given, $z = px + qy$ where $p,q > 0$ such that the maximum of $z$ occurs at both the points (15,15) and (0,20) where the corner points of the feasible region are (0,10), (5,5), (15,15), (0,20).
Let us consider ${z_{\max }}$ to be the maximum value of $z$ in the feasible region.
Since maximum occurs at both (15,15) and (0,20)
Therefore, ${z_{\max }}$ is attained at both the points
$\Rightarrow {z_{\max }} = p(15) + q(15)$and ${z_{\max }} = p(0) + q(20)$

$$\Rightarrow p(15) + q(15) = p(0) + q(20) \\\ \Rightarrow 15p + 15q = 20q \\\ \Rightarrow 15p = 5q \\\ \Rightarrow 3p = q \\\$$

Therefore, option D. $q = 3p$ is the required solution.

Note: Observe that the maximum value attained by $z$ at any point on the feasible region is always the same.