Maths. Section ALinear Programming

The feasible region represented by the constraints: x + 2y ≥ 100,\ 2x…

The feasible region represented by the constraints: $x + 2y \ge 100,\ 2x - y \le 0,\ 2x + y \le 200,\ x \ge 0,\ y \ge 0$ of an LPP is:

$Graph of the regions for the LPP$

Answer & explanation

Correct answer: option 3

The correct answer is Option (3) → Region C

Each constraint is a half-plane; the side of each boundary line is found with the origin test.

$x + 2y \ge 100$: origin gives $0 \ge 100$, false, so the side away from the origin.

$2x - y \le 0$, that is $y \ge 2x$: the side above the line $y = 2x$, towards the y-axis.

$2x + y \le 200$: origin gives $0 \le 200$, true, so the side containing the origin.

$x \ge 0,\ y \ge 0$: the first quadrant.

The corner points of the common region are where these lines meet:

    $x = 0$ with $x + 2y = 100$: (0, 50)
    $y = 2x$ with $x + 2y = 100$: (20, 40)
    $y = 2x$ with $2x + y = 200$: (50, 100)
    $x = 0$ with $2x + y = 200$: (0, 200)

So the feasible region lies between the y-axis and the line $y = 2x$, above $x + 2y = 100$ and below $2x + y = 200$. For example, (10, 100) satisfies every constraint:

$10 + 200 = 210 \ge 100,\quad 20 - 100 = -80 \le 0,\quad 20 + 100 = 120 \le 200$

This is Region C.