Question

If $Z=5x+10y,$ subject to the constraints $x+2y\le 120,$ $x+y\ge 60,$ $x-2y\ge 0$ and $x\ge 0,y\ge 0,$ then the maximum value of $Z$ occurs at

[1 mark]

• A. only one point
• B. only two points
• C. more than two points but finite
• D. infinite number of points

Answer 1

The correct option is D infinite number of points

$Z=5x+10y$
subject to the constraints
$x+2y\le 120,x+y\ge 60,x-2y\ge 0$
and $x\ge 0,y\ge 0$
We draw the given inequalities to find the feasible region bounded by these constraints.


The shaded region $BDEF$ determined by linear inequalities shows the feasible region.
Let us evaluate the objective function $Z$ at each corner point.

Corner point	$Z=5x+10y$
$B(120,0)$	$600$
$D(60,0)$	$300$
$E(40,20)$	$400$
$F(60,30)$	$600$

We can observe, the maximum value of $Z$ is $600$ at $F(60,30)$ and $B(120,0).$
Hence, $Z$ attains maximum value at every point on the line segment joining $B(120,0)$ and $F(60,30).$