A school canteen makes x veg sandwiches and y cheese sandwiches in a morning. Each sandwich uses 2 slices of bread, and 100 slices of bread are available. Each cheese sandwich also uses 1 slice of cheese, and 30 slices of cheese are available. A veg sandwich takes 2 minutes to make and a cheese sandwich takes 1 minute, and 80 minutes are available. The profit is Rs. 6 on a veg sandwich and Rs. 8 on a cheese sandwich, and all the sandwiches are sold. Consider the following statements:
(A) The time constraint is \(2x + y \le 80\).
(B) \((30, 20)\) is a corner point of the feasible region.
(C) The maximum profit is Rs. 360, made with 20 veg and 30 cheese sandwiches.
(D) When the profit is maximum, all the 80 minutes are used.
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → (A), (B) and (C) only
The linear programming problem is
Maximise \(Z = 6x + 8y\)
subject to
Bread: \(2x + 2y \le 100\), that is, \(x + y \le 50\)
Cheese: \(y \le 30\)
Time: \(2x + y \le 80\)
\(x \ge 0, \; y \ge 0\)
Meeting points of the boundary lines:
\(2x + y = 80\) and \(x + y = 50\): \(x = 30, \; y = 20\)
\(x + y = 50\) and \(y = 30\): \(x = 20\)
The corner points are (0, 0), (40, 0), (30, 20), (20, 30) and (0, 30). Values of Z:
\(Z(40, 0) = 240\)
\(Z(30, 20) = 180 + 160 = 340\)
\(Z(20, 30) = 120 + 240 = 360\)
\(Z(0, 30) = 240\)
So the time constraint is \(2x + y \le 80\), (30, 20) is a corner point, and the maximum profit is Rs. 360 at (20, 30), where the time used is \(2(20) + 30 = 70\) minutes.
Checking the statements
(A) The time constraint is \(2x + y \le 80\). — Correct. 2 minutes for each veg and 1 minute for each cheese sandwich, within 80 minutes.
(B) \((30, 20)\) is a corner point of the feasible region. — Correct. It is where \(2x + y = 80\) meets \(x + y = 50\).
(C) The maximum profit is Rs. 360, made with 20 veg and 30 cheese sandwiches. — Correct. \(Z(20, 30) = 360\) is the largest corner value.
(D) When the profit is maximum, all the 80 minutes are used. — Incorrect. At (20, 30) only 70 minutes are used.