Which of the region shown in the given figures represents the feasible region bounded by the following constraints?

$4x+y≥80,2x+y≥60,x+y≤80,x≥0,y≥0$

region C

The correct answer is Option (3) → region C

$ \text{We test the origin (0, 0) for each inequality to find the correct region.} $

$ \text{Constraints are: }x\ge0\text{ and }y\ge0.\text{ This restricts the region to the first quadrant.} $

$ \text{1. }x+y\le80:\text{ Test (0, 0) }⇒0+0\le80\text{ (True). The region is below the line passing through (80, 0) and (0, 80).} $

$ \text{2. }4x+y\ge80:\text{ Test (0, 0) }⇒0+0\ge80\text{ (False). The region is above the line passing through (20, 0) and (0, 80).} $

$ \text{3. }2x+y\ge60:\text{ Test (0, 0) }⇒0+0\ge60\text{ (False). The region is above the line passing through (30, 0) and (0, 60).} $

$ \text{The feasible region must be in the first quadrant, below line 1, above line 2, and above line 3.} $

Region D is above line 3 (2x + y ≥ 60) but below line 2 (4x + y ≥ 80). Therefore, it does not satisfy the condition 4x + y ≥ 80 and hence it is not part of the feasible region.

$ \text{Region B is below both line 2 and line 3. This is incorrect.} $

$ \text{Region A is below line 2, but above line 3. This is incorrect.} $

$ \text{Region C is below line 1 (x+y=80), above line 2 (4x+y=80), and above line 3 (2x+y=60). } $

$ \text{Therefore, Region C represents the feasible region.} $