 Hi and how are you all today? The question says the manufacturer makes two types of cups and A and B. Three machines are required to manufacture the cups and the time in minutes required by each is given to us in this table. Each machine is available for a maximum period of 6 hours per day. If the profit on each cup is 75 paisa and on B is 50 paisa, show that 15 cups of type A and 30 cups of type B should be manufactured per day to get the maximum profit. So let us formulate this question. First of all, let X cups of type A be there and Y cups of type B be there. We need to maximize the profit. Let us take it as Z that is 0.75X plus 0.50Y subject to the constraints. We are given that each machine is available for a maximum period of 6 hours. So that means 12X plus 6Y is less than equal to 6R that is equal to 360 minutes. This further implies that 2X plus Y should be less than equal to 60. Similarly the other will be 18X plus 0Y should be less than equal to 360. This implies that the value of X should be less than equal to 20. And further we have 6X plus 9Y should be less than equal to 360 that further implies 2X plus 3Y should be less than equal to 120. And obviously both X and Y should be greater than equal to 0. Now let us find out the two points for each of these inequalities. Let us first make these inequalities an equation. So we have to take this inequality as 2X plus Y is equal to 60 for that we have 2 points 0, 60, 0, 30. Then we need to take 2X plus 3Y equal to 120 then the two points are then X is 0, Y is 40 and Y is 0 then X is 60. And X is equal to 20, X is greater than 0 and Y is also greater than equal to 0. So let us plot these three lines on the graph. Now we have plotted one line that is 2X plus Y is equal to 60. In the same way we have X is equal to 20. So it will be that is X is equal to 20. And the third line is when X is 0, Y is 40. So this will be this point and when Y is 0, X is 60. So we have the required line like this. This is representing 2X plus 3Y equal to 120. Now the feasible region will be the shaded region with end points as 0, 0 as it is included in each of the inequalities. Then we have point A that is 20, 0. Then we have point B, 20, 20. Then we have point C as having coordinates 15, 30. Then we have D having coordinates 0, 40. Now on all these points that is O, A, B, C and D we will find out the value of Z. Now let us find out we have here Z1 as 0 plus 0 which gives us the net value of Z as 0. Then we have 0.75 into 20 plus 0 into 0. That gives us the value of Z2 as rupees 15. Similarly we will find out Z3, Z4 and Z5. Now we have Z3, Z4 and Z5's net value as rupees 25, rupees 26.25 and then rupees 20. So we can conclude here that the maximum profit is rupees 26.25. When maximum profit is rupees 26.25 when cups of type A are 15 and cups of type B is equal to 30. And this is what we needed to prove in this question. So hence we have proved the given question. So this completes the session. Hope you understood the whole concept well and enjoyed it too. Have a nice day.