The corner points of the bounded feasible region of the LPP: Maximize \(Z = x + y\) subject to constraints \(2x + 5y \le 100\), \(8x + 5y \le 200\), \(x \ge 0, y \ge 0\) are
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → \((0, 0), (25, 0), (0, 20), \left(\displaystyle\frac{50}{3}, \frac{40}{3}\right)\)
The intercepts of the two boundary lines are:
\(2x + 5y = 100\): \((50, 0)\) and \((0, 20)\)
\(8x + 5y = 200\): \((25, 0)\) and \((0, 40)\)
On the x-axis the tighter limit is \(x \le 25\) (from \(8x + 5y \le 200\)); on the y-axis it is \(y \le 20\) (from \(2x + 5y \le 100\)). So \((50, 0)\) and \((0, 40)\) are not feasible:
\((50, 0)\): \(8(50) = 400 \gt 200\)
\((0, 40)\): \(5(40) = 200 \gt 100\)
The lines meet where, subtracting the first from the second,
\(6x = 100\)
\( \displaystyle x = \frac{50}{3} \)
\( \displaystyle 5y = 100 - \frac{100}{3} = \frac{200}{3},\quad y = \frac{40}{3} \)
The feasible region is the quadrilateral with corner points
\( \displaystyle (0, 0),\ (25, 0),\ \left(\frac{50}{3}, \frac{40}{3}\right),\ (0, 20) \)