The feasible region represented by the constraints 4x+y ≥ 80, x+5y ≥ 115, 3x +2y ≤150, x,y ≥ 0 of an LPP is:

Correct Answer: Region C

To determine which of the labeled regions (Region A, Region B, Region C, or Region D) represents the feasible region for the given Linear Programming Problem (LPP), we analyze each constraint one by one by identifying its boundary line and corresponding half-plane in the provided graph.

1. Constraint 1:
$4x+y\ge 80$
The boundary line is $4x+y=80$, which passes through the points $(0,80)$ and $(20,0)$ as labeled on the axes.
Testing the origin $(0,0)$:
$4\left(0\right)+0=0\ngeqq 80$ (False).
Therefore, the feasible region lies on or above/to the right of this line (away from the origin). This excludes Region D and Region A, which lie to the left/below this line.

2. Constraint 2:
$x+5y\ge 115$
The boundary line is $x+5y=115$, which passes through the points $(0,23)$ and $(115,0)$ as shown in the graph.
Testing the origin $(0,0)$:
$0+5\left(0\right)=0\ngeqq 115$ (False).
Thus, the feasible region must lie on or above this boundary line (away from the origin). This excludes Region B and Region A, which lie below this line.

3. Constraint 3:
$3x+2y\le 150$
The boundary line is $3x+2y=150$, which passes through the points $(0,75)$ and $(50,0)$ on the graph.
Testing the origin $(0,0)$:
$3\left(0\right)+2\left(0\right)=0\le 150$ (True).
Thus, the feasible region must lie below or to the left of this boundary line (towards the origin).

4. Non-negativity Constraints:
$x\ge 0,y\ge 0$ restrict the region to the first quadrant.

Intersection of All Regions:
Taking the intersection of the half-planes:
- Above $4x+y=80$ (excludes A and D)
- Above $x+5y=115$ (excludes A and B)
- Below $3x+2y=150$
This leaves the shaded Region C (bounded by the corner points $(2,72)$, $(15,20)$, and $(40,15)$) as the only region satisfying all constraints simultaneously.