A factory makes tennis rackets and cricket bats. A tennis racket takes 1.5 hours of machine time and 3 hours of craftsman's time in its making while a cricket bat takes 3 hours of machine time and 1 hour of craftsman'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. If the profit on a racket and on a bat is Rs. 20 and Rs. 10 respectively, find the number of tennis rackets and cricket bats that the factory must manufacture to earn the maximum profit. Make it as an L.P.P. and solve graphically.
Answer: Max profit = Rs 200 at (4, 12): 4 rackets and 12 bats
Solution
Let the factory make x rackets and y bats per day. Profit: Z=20x+10y. Machine time: 1.5x+3y≤42⟹x+2y≤28 Craftsman time: 3x+y≤24 x≥0,y≥0 Corner points: (0,0), (8,0), (4,12) (intersection of x+2y=28 and 3x+y=24) and (0,14). Evaluate Z=20x+10y: (8,0): Z=160 (0,14): Z=140 (4,12): Z=80+120=200 ← maximum
∴Zmax=Rs 200 at (4,12)
The factory should make 4 tennis rackets and 12 cricket bats.Feasible region for 1.5x + 3y = 42 and 3x + y = 24, with corner points O(0, 0), A(8, 0), C(4, 12) and B(0, 14).
