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’s time in its making while a cricket bat takes 3 hour of machine time and 1 hour of craft man’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. (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's 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’s 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.