 Hi and how are you all today? The question says, a firm makes item A and V and the total number of items it can make in a day is 24. It takes 1 hour to make an item of A and only half an hour to make an item of B. The maximum time available per day is 16 hours. The profit on item A is 300 and on one item of B is 160. How many items of each type should be produced to maximize the profits? Solve the problem graphically. So here first of all, let item A manufactures per day BX. Item B manufactured per day BY. Now the constraints are, first of all it is given to us that maximum time available to us or sorry the total number of item which we can make in a day is 24. That means X plus 5 can be less than equal to 24. Right? It has to be that X and Y has to be greater than 0. Then we are given that it takes 1 hour to make item X and half an hour to make item Y. Right? And the total time should be less than equal to 16 or we can write this equation as 2X plus Y is less than equal to and we need to maximize our profit function, say P. Profit earned on X is 300 and profit earned on Y is 160. So we need to maximize this profit function with respect subject to these constraints. Now let's take X plus Y equal to 24, 2X plus Y equal to 32, X greater than equal to 0 and Y greater than equal to 0 separately. Now here we will be finding out two points for these equations. The value of X is 0, Y is 24, Y is 0, X is 24. For this equation we have when the value of X is 0, Y is 32 and when the value of Y is 0 then the value of X is 16. So now we need to plot these two equations on the graph. Now here since the value of X and Y are both greater than equal to 0, so we will be concentrating on the first quadrant itself. Now let us plot those two equations here. Now here are the two equations and these are the lines representing these two equations. They both are intersecting at point P where the value of X is coming out to be 8 and Y is coming out to be 16. So we can write that the region will contain the point 0, 0 in both the points. So we have point 0 then A that is 16, 0 then point B that is 8 comma 16 and point C that is 0 comma 24. Now at these four points let us find out the value of P. So we have it as equal to 300 into 0 plus 160 into 0 giving us the value of P as 0. Then here we have 300 into 16 plus 160 into 0 which gives us the value as 4,800. Here we have into 8 plus 160 into 60 and the value is coming out to be 4,960 and lastly we have 300 into 0 plus 160 into 24 giving us the value as 3,840. So we can clearly see that profit is maximum that is 4,960 when X is equal to 8 and Y is equal to 60. Hence 8 items of A and 16 items of B per day should be manufactured. Right, so this completes the session. Hope you understood the whole concept well and enjoy. Bye for now.