A water-treatment plant uses x kg of chemical P and y kg of chemical Q a day. Each kg of P gives 3 units of disinfectant and 1 unit of coagulant, and each kg of Q gives 1 unit of disinfectant and 2 units of coagulant. At least 15 units of disinfectant and at least 10 units of coagulant are needed a day. Chemical P costs Rs. 40 per kg and chemical Q costs Rs. 30 per kg. Which of the following statements are correct?
(A) The feasible region is unbounded.
(B) (4, 3) is a corner point of the feasible region.
(C) The minimum daily cost is Rs. 250.
(D) The largest value of the cost on the feasible region is Rs. 450.
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → (A), (B) and (C) only
Let x kg of P and y kg of Q be used. Then
Minimise \(Z = 40x + 30y\)
subject to
\(3x + y \ge 15\) (disinfectant)
\(x + 2y \ge 10\) (coagulant)
\(x \ge 0,\ y \ge 0\)
The two lines meet where, putting \(x = 10 - 2y\),
\(30 - 6y + y = 15\)
\(y = 3,\ x = 4\)
The feasible region lies above both lines and goes on without end, so it is unbounded, with corner points (0, 15), (4, 3) and (10, 0). Values of Z:
\(Z(0, 15) = 450\)
\(Z(4, 3) = 160 + 90 = 250\)
\(Z(10, 0) = 400\)
The half-plane \(40x + 30y \lt 250\) has no point in common with the region, so the minimum is 250. Large values of x or y keep the point feasible and make Z as large as we like, so Z has no largest value.
So the region is unbounded, (4, 3) is a corner, the minimum cost is Rs. 250 and there is no largest cost.
Checking the statements
(A) The feasible region is unbounded. — Correct. Both requirements are “at least” constraints, so the region goes on without end.
(B) (4, 3) is a corner point of the feasible region. — Correct. It is where \(3x + y = 15\) meets \(x + 2y = 10\).
(C) The minimum daily cost is Rs. 250. — Correct. \(Z(4, 3) = 250\), and \(40x + 30y \lt 250\) has no feasible point.
(D) The largest value of the cost on the feasible region is Rs. 450. — Incorrect. (0, 20) is feasible and costs Rs. 600, so the cost has no largest value.