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

Let the number of rackets and the number of bats to be made be x and y respectively.

The machine time available is not more than 42 hours.

∴ 1.5x +3y ≤ 42

The craftsman`s time is not available for more than 24 hours

∴ 3x + y ≤ 24

The factory is to work at full capacity

∴ 1.5x + 3y = 42

3x + y = 24

Where x and y ≥ 0

Solving the two equations we get x= 4 and y = 12

Thus 4 rackets and 12 bats must be made

(i) The given information can be complied in the tables form as follows

The profit on a racket is Rs 20 and on bat is Rs 10

Maximize, Z = 20x +10y ………………1

Subject to constraints

1.5x + 3y = 42

3x + y = 24

x, y ≥ 0

the feasible region determined by the system of constraints is:

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:

Thus the maximum profit of the factory is when it works to its full capacity is Rs200.