A school needs seating for at least 60 students in a hall. A long bench seats 3 students and costs Rs. 800, and a short bench seats 2 students and costs Rs. 500. The hall has room for at most 25 benches. The least cost of the benches is
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → Rs. 15500
Let x long benches and y short benches be bought. Then
Minimise \(Z = 800x + 500y\)
subject to
\(3x + 2y \ge 60\)
\(x + y \le 25\)
\(x \ge 0, \; y \ge 0\)
The lines \(3x + 2y = 60\) and \(x + y = 25\) meet where
\(3x + 2(25 - x) = 60\)
\(x = 10, \; y = 15\)
The feasible region is the triangle with corner points (10, 15), (20, 0) and (25, 0). Values of Z:
\(Z(10, 15) = 8000 + 7500 = 15500\)
\(Z(20, 0) = 16000\)
\(Z(25, 0) = 20000\)
So the least cost is Rs. 15500, with 10 long and 15 short benches.
Why the other options are incorrect
Option (1) Rs. 15000 — Incorrect. This needs 30 short benches, but the hall holds at most 25 benches.
Option (3) Rs. 16000 — Incorrect. This is the cost of 20 long benches only, which is more than at (10, 15).
Option (4) Rs. 20000 — Incorrect. This is the largest corner value, at (25, 0).