(i) Let the number of rackets and the number of bats to be made be x and y respectively.The machine time is not available for more than 42 hours.∴1.5x+3y≤42 .....(1)The craftsman's time is not available for more than 24 hours.∴3x+y≤24 .....(2)The factory is to work at full capacity. Therefore,1.5x+3y=42 3x+y=24 On solving these equations, we obtain x = 4 and y = 12Thus, 4 rackets and 12 bats must be made.(i) The given information can be complied in a table as follows.∴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 = 20 x+10 ...... (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.Thus, the maximum profit of the factory when it works to its full capacity is Rs 200 .