In the given figure, feasible region represented by the constraints 4…

The correct answer is Option (3) → Region C

Given constraints

$4x+y\ge80$

$x+5y\ge115$

$3x+2y\le150$

$x\ge0,\;y\ge0$

Find intersection points of boundary lines

Intersection of $4x+y=80$ and $x+5y=115$

$y=80-4x$

$x+5(80-4x)=115$

$x+400-20x=115$

$-19x=-285$

$x=15$

$y=20$

Point $(15,20)$

Intersection of $4x+y=80$ and $3x+2y=150$

$y=80-4x$

$3x+2(80-4x)=150$

$3x+160-8x=150$

$-5x=-10$

$x=2$

$y=72$

Point $(2,72)$

Intersection of $x+5y=115$ and $3x+2y=150$

$x=115-5y$

$3(115-5y)+2y=150$

$345-15y+2y=150$

$-13y=-195$

$y=15$

$x=40$

Point $(40,15)$

Intersection with axes

From $4x+y=80$, at $x=0$, $y=80$ but does not satisfy $x+5y\ge115$

From $x+5y=115$, at $y=0$, $x=115$ but violates $3x+2y\le150$

From $3x+2y=150$, at $x=0$, $y=75$ but violates $x+5y\ge115$

Hence feasible corner points are

$(2,72),\;(15,20),\;(40,15)$

The feasible region is the triangular region with vertices $(2,72),\;(15,20),\;(40,15)$.