Consider the feasible region determined by the constraints \(x \le 4\), \(y \le 3\), \(x \ge 0\), \(y \ge 0\).
(A) The feasible region is a rectangle
(B) (4, 3) is a corner point of the feasible region
(C) The feasible region is unbounded
(D) The maximum value of \(Z = 2x + y\) on this region is 11
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → (A), (B) and (D) only
The four constraints keep x between 0 and 4 and y between 0 and 3. The region is enclosed by the lines x = 0, x = 4, y = 0 and y = 3, with corner points (0, 0), (4, 0), (4, 3) and (0, 3).
Values of Z = 2x + y at the corners:
\(Z(0, 0) = 0\)
\(Z(4, 0) = 8\)
\(Z(4, 3) = 11\)
\(Z(0, 3) = 3\)
So the region is a bounded rectangle with corners (0, 0), (4, 0), (4, 3), (0, 3), and the maximum of Z is 11.
Checking the statements
(A) The feasible region is a rectangle — Correct. Its sides lie on x = 0, x = 4, y = 0 and y = 3.
(B) (4, 3) is a corner point of the feasible region — Correct. The lines x = 4 and y = 3 meet there.
(C) The feasible region is unbounded — Incorrect. Both x and y are limited, so the region is bounded.
(D) The maximum value of \(Z = 2x + y\) on this region is 11 — Correct. \(Z(4, 3) = 8 + 3 = 11\), the largest corner value.