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

The correct answer is Option (3) → Region C

The origin O(0, 0) decides the side of each line:

    \(4x + y \ge 80\): O gives \(0 \ge 80\), false, so the region is on the side away from O.
\(x + 5y \ge 115\): O gives \(0 \ge 115\), false, so the region is on the side away from O.
\(3x + 2y \le 150\): O gives \(0 \le 150\), true, so the region is on the side of O.

The corners where the lines meet:

    \(4x + y = 80\) and \(x + 5y = 115\): \(x + 5(80 - 4x) = 115\), so \(x = 15,\ y = 20\)
\(4x + y = 80\) and \(3x + 2y = 150\): \(3x + 2(80 - 4x) = 150\), so \(x = 2,\ y = 72\)
\(x + 5y = 115\) and \(3x + 2y = 150\): \(3(115 - 5y) + 2y = 150\), so \(y = 15,\ x = 40\)

So the feasible region is the triangle with corners (2, 72), (15, 20) and (40, 15), which lies above the first two lines and below the third. This is Region C.

As a check, the point (19, 36) inside this triangle satisfies all the constraints:

    \(4(19) + 36 = 112 \ge 80\)
\(19 + 5(36) = 199 \ge 115\)
\(3(19) + 2(36) = 129 \le 150\)