Question -
Answer -
Let x and y be the number of rackets and the number of bats tobe made.
Given that the machine time is not available for more than 42hours
Hence, 1.5x + 3y ≤ 42 ……………. (i)
Also, given that the craftsman’s time is not available for morethan 24 hours
Hence, 3x + y ≤ 24 ………… (ii)
The factory is to work at full capacity. Hence,
1.5x + 3y = 42
3x + y = 24
On solving these equations, we get
x = 4 and y = 12
Therefore, 4 rackets and 12 bats must be made.
(i) The given information can be compiled in a table as givebelow
| 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
Since, the profit on a racket is Rs 20 and Rs 10
Hence, Z = 20x + 10y
The mathematical formulation of the given problem can be writtenas
Maximize Z = 20x + 10y ………….. (i)
Subject to the constraints,
1.5x + 3y ≤ 42 …………. (ii)
3x + y ≤ 24 …………….. (iii)
x, y ≥ 0 ………………… (iv)
The feasible region determined by the system of constraints isgiven below
A (8, 0), B (4, 12), C (0, 14) and O (0, 0) are the cornerpoints respectively.
The values of Z at these corner points are given below
| Corner point | Z = 20x + 10y | |
| A (8, 0) | 160 | |
| B (4, 12) | 200 | Maximum |
| C (0, 14) | 140 | |
Therefore, the maximum profit of the factory when it works toits full capacity is Rs 200