Question

The corner points of the feasible region determined by the following system linear inequalities:
$2x+y\le 10,x+3y\le 15,xy\ge 0$ are $(0,0),(5,0),(3,4)$ and $(0,5)$.
Let $Z=px+qy$, where $p,q>0$. Condition on $p$ and $q$ so that the maximum of $Z$ occurs at both $(3,4)$ and $(0,5)$ is

• A. $p=q$
• B. $p=2q$
• C. $p=3q$
• D. $q=3p$

Answer 1

The correct option is D $q=3p$

The maximum value of $Z$ is unique.
It is given that the maximum value of $Z$ occurs at two points, $(3,4)$ and $(0,5)$
$\therefore$ Value of $Z$ at $(3,4)=$ Value of $Z$ at $(0,5)$
$\Rightarrow$ $p\left(3\right)+q\left(4\right)=p\left(0\right)+q\left(5\right)$
$\Rightarrow$ $3p+4q=5q\Rightarrow$ $q=3p$