The corner points of the feasible region determined by a system of linear constraints are (0, 0), (0, 40), (20, 40), (60, 20), (60, 0). If the objective function is $Z=4x+3y$, then which one of the following is true?
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → $\text{Max. Z < 400}$
Given objective function: $Z = 4x + 3y$
Corner points:
(0, 0), (0, 40), (20, 40), (60, 20), (60, 0)
Compute $Z$ at each corner point:
At (0, 0): $Z = 4(0) + 3(0) = 0$
At (0, 40): $Z = 4(0) + 3(40) = 120$
At (20, 40): $Z = 4(20) + 3(40) = 80 + 120 = 200$
At (60, 20): $Z = 4(60) + 3(20) = 240 + 60 = 300$
At (60, 0): $Z = 4(60) + 3(0) = 240$
Maximum value: $Z_{\max} = 300$ at $(60, 20)$
Minimum value: $Z_{\min} = 0$ at $(0, 0)$
Therefore, $Z_{\max} = 300$ at $(60, 20)$ and $Z_{\min} = 0$ at $(0, 0)$.