Consider the feasible region determined by the constraints \(x + 2y \le 10\), \(3x + y \le 15\), \(x \ge 0\), \(y \ge 0\).
Match each objective function in List-I with its maximum value on this region in List-II.
| List-I | List-II | ||
| (A) | \(Z = x + y\) | (I) | 20 |
| (B) | \(Z = 2x + y\) | (II) | 7 |
| (C) | \(Z = x + 4y\) | (III) | 25 |
| (D) | \(Z = 5x + y\) | (IV) | 11 |
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → (A)-(II), (B)-(IV), (C)-(I), (D)-(III)
The region is bounded, so each maximum is at a corner point. The line 3x + y = 15 meets the x-axis at (5, 0), and x + 2y = 10 meets the y-axis at (0, 5). The two lines meet where
\(y = 15 - 3x\)
\(x + 2(15 - 3x) = 10\)
\(30 - 5x = 10\)
\(x = 4,\ y = 3\)
The corner points are (0, 0), (5, 0), (4, 3) and (0, 5). At (0, 0) every Z is 0.
(A) \(Z = x + y\)
This matches (II) 7.
\(Z(5, 0) = 5,\ Z(4, 3) = 7,\ Z(0, 5) = 5\)
(B) \(Z = 2x + y\)
This matches (IV) 11.
\(Z(5, 0) = 10,\ Z(4, 3) = 11,\ Z(0, 5) = 5\)
(C) \(Z = x + 4y\)
This matches (I) 20.
\(Z(5, 0) = 5,\ Z(4, 3) = 16,\ Z(0, 5) = 20\)
(D) \(Z = 5x + y\)
This matches (III) 25.
\(Z(5, 0) = 25,\ Z(4, 3) = 23,\ Z(0, 5) = 5\)