A linear programming problem is as follows : Minimize Z=30 x+50 y… - Vidyadip

MCQ

A linear programming problem is as follows : Minimize $Z=30 x+50 y$ Subject to the constraints, $3 x+5 y \geq 15\ ,\ 2 x+3 y \leq 18 \ ,\ x \geq 0, y \geq 0$ In the feasible region, the minimum value of $Z$ occurs at

A
a unique point
B
no point
✓
infinitely many points
D
two points only

✓

Answer

Correct option: C.

infinitely many points

Here, the feasible region is shaded.

Corner points	Value of $Z=30 x+50 y$
$A(0,3)$	$30 xx0+50 xx3=150 ($Minimum$)$
$B(5,0)$	$30 xx5+50 xx0=150 ($Minimum$)$
$C(9,0)$	$30 xx9+50 xx0=270$
$D(0,6)$	$30 xx0+50 xx6=300$

Since, minimum value of $Z$ occurs at both $A$ and $B$.
So, $Z$ is minimum at every point on the line joining $A B$.
So, minimum value of $Z$ occurs at infinitely many points.

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