A workshop has to make at least 30 lamps in a week, x of type A and y of type B. A lamp of type A takes 2 hours and a lamp of type B takes 3 hours, and at most 50 hours are available in the week. The profit is Rs. 100 on a lamp of type A and Rs. 150 on a lamp of type B. If the weekly profit is to be maximised, then
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → the problem has no feasible solution
The constraints are
\(x + y \ge 30\) (order)
\(2x + 3y \le 50\) (time)
\(x \ge 0,\ y \ge 0\)
Every lamp takes at least 2 hours, so for any non-negative x and y
\(2x + 3y \ge 2x + 2y = 2(x + y)\)
If \(x + y \ge 30\), this gives \(2x + 3y \ge 60 \gt 50\). No point satisfies both constraints, so the feasible region is empty and the problem has no feasible solution.
Why the other options are incorrect
Option (1) the maximum profit is Rs. 2,500 — Incorrect. Rs. 2,500 is the profit when all 50 hours are used, but then at most 25 lamps are made, fewer than 30.
Option (2) the maximum profit is Rs. 3,000 — Incorrect. No plan satisfies both constraints, so no profit value is attained.
Option (4) the profit has no maximum value, as the feasible region is unbounded — Incorrect. The feasible region is empty, not unbounded.