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

Hint: 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.