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प्रश्न
A factory makes tennis rackets and cricket bats. A tennis racket takes 1.5 hours of machine time and 3 hours of craftsman’s time in its making while a cricket bat takes 3 hour of machine time and 1 hour of craftsman’s time. In a day, the factory has the availability of not more than 42 hours of machine time and 24 hours of craftsman’s time.
If the profit on a racket and on a bat is Rs 20 and Rs 10 respectively, find the maximum profit of the factory when it works at full capacity.
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उत्तर
The given information can be complied in a table as follows.
| Tennis Racket | Cricket Bat | Availability | |
| Machine Time (h) | 1.5 | 3 | 42 |
| Craftsman’s Time (h) | 3 | 1 | 24 |
∴ 1.5x + 3y ≤ 42
3x + y ≤ 24
x, y ≥ 0
The profit on a racket is Rs 20 and on a bat is Rs 10.
`:. Z =20x + 10y`
The mathematical formulation of the given problem is
Maximize Z =20x + 10y … (1)
subject to the constraints,
1.5x + 3y ≤ 42 … (2)
3x + y ≤ 24 … (3)
x, y ≥ 0 … (4)
The feasible region determined by the system of constraints is as follows.
The corner points are A (8, 0), B (4, 12), C (0, 14), and O (0, 0).
The values of Z at these corner points are as follows.
| Corner point | Z = 20x + 10y | |
| A(8, 0) | 160 | |
| B(4, 12) | 200 | → Maximum |
| C(0, 14) | 140 | |
| O(0, 0) | 0 |
Thus, the maximum profit of the factory when it works to its full capacity is Rs 200.
