Question

The corner points of the feasible region determined by a system of linear constraints are $(0,10),(5,5),(15,15),(0,20).$ Let $z=px+qy$ where $p,q>0.$ If the maximum of $z$ occurs at both the points $(15,15)$ and $(0,20),$ then which of the following is true:

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

Answer 1

The correct option is D $q=3p$

Let $z_0$ be the maximum value of $z$ in the feasible region. Since maximum occurs at both $(15,15)$ and $(0,20)$, the value $z_0$​ is attained at both $(15,15)$ and $(0,20)$.
$\Rightarrow z_0=p\left(15\right)+q\left(15\right)\cdots \left(i\right)$
and $z_0=p\left(0\right)+q\left(20\right)\cdots \left(ii\right)$
From $\left(i\right)$ and $\left(ii\right),$ we get:
$15p+15q=20q$
$\Rightarrow 15p=5q$
$\Rightarrow 3p=q$