Question

A factory makes tennis rackets and cricket bats. A tennis racket takes 1.5 hours of machine time and 3 hours of craft man’ time in its making while a cricket bat takes 3 hour of machine time and 1 hour of craft man’ time.

In a day, the factory has the availability of not more than 42 hours of machine time and 24 hours of craftsman’* time. (i) What number of rackets and bats must be made if the factory is to work at full capacity? (ii) 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.

Answer 1

Let the number of rackets and the number of cricket bats to be made in a day be x and y respectively.

Construct the following table:

| Item | Number | Machine time (in h) | *Craftsman' time (in h) | Profit (in Rs.) |
| Tennis rackets | x | 1.5x | 3x | 20x |
| Cricket bats | y | 3y | 1y | 10y |
| Total | x + y | 1.5x + 3y | 3x + y | 20x + 10y |
| Availability** | | 42 | 24 | |

The machine time is not available for more than 42 h.

∴ 1.5x + 3y ≤ 42

The craftman’* time is not available for more than 24 h.

∴ 3x + y ≤ 24

The profit on rackets is Rs. 20 and on bats is Rs. 10.

∴Maximum Z = 20x + 10y …(i)

Subject to constraints 1.5x + 3y ≤ 42 …(ii)

3x + y ≤ 24 …(iii)

x ≥ 0, y ≥ 0 …(iv)

Firstly, draw the graph of the line 1.5x + 3y = 42

Secondly, draw the graph of the line 3x + y = 24

On solving equations 1.5x + 3y = 42 and 3x + y = 24, we get B(4, 12).

∴ Feasible region is OABCO( See below figure).

The corner points of the feasible region are O (0, 0), A(8, 0), B(4, 12) and C (0, 14). The values of Z at these points are as follows:

Thus, the maximum profit of the factory when it works to its full capacity is Rs. 200.