A housing society needs at least 24,000 litres of water a day from tankers. A trip of tanker A brings 6,000 litres and costs Rs. 900, and a trip of tanker B brings 4,000 litres and costs Rs. 500. If x trips of A and y trips of B are made, the least daily cost is
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → Rs. 3,000
The problem is
Minimise \(Z = 900x + 500y\)
subject to
\(6000x + 4000y \ge 24000\), that is \(3x + 2y \ge 12\)
\(x \ge 0,\ y \ge 0\)
The feasible region is unbounded, with corner points (4, 0) and (0, 6). Values of Z:
\(Z(4, 0) = 3600\)
\(Z(0, 6) = 3000\)
The half-plane \(900x + 500y \lt 3000\) has no point in common with the region: on the line \(3x + 2y = 12\), \(y = 6 - 1.5x\) and
\(Z = 900x + 3000 - 750x = 3000 + 150x \ge 3000\)
So the least cost is Rs. 3,000, with 6 trips of tanker B.
Why the other options are incorrect
Option (2) Rs. 3,300 — Incorrect. This is the cost of 2 trips of A and 3 of B, a feasible plan but not the cheapest.
Option (3) Rs. 3,600 — Incorrect. This is the cost at the other corner, 4 trips of A.
Option (4) Rs. 2,700 — Incorrect. 3 trips of A bring only 18,000 litres, less than 24,000.