 Hi and welcome to the session. Let's work out the following question. The question says a house pipe wishes to mix up two kinds of foods X and Y in such a way that mixture contains at least 10 units of vitamin A, 12 units of vitamin B and 8 units of vitamin C. The vitamin contains of 1 kg of foods X and 1 kg of food Y are as given in the following table. Now if 1 kg of food X costs rupees 6 and 1 kg of food Y costs rupees 10, find the least cost of the mixture which will produce the desired diet. So let us see the solution to this question. First of all let X kg of food X and Y kg of food Y be mixed to get the desired diet. Then the LPP is to minimize Z equal to 6X plus 10Y because the profit on each curve A, sorry because if 1 kg of food X is costing rupees 6 and 1 kg of food Y is costing rupees 10, so we have to minimize Z that is equal to 6X plus 10Y. Now the constraints will be X into 1 plus 2 into Y is greater than equal to 10 for vitamin A, 2 into X plus 2 into Y should be greater than equal to 12 for vitamin B and 3 into X plus 1 into Y should be greater than equal to 8 for vitamin C. So let us just write down these constraints. So the LPP is to minimize Z equal to this, subject to the constraints X plus 2Y greater than equal to 10, 2X plus 2Y greater than equal to 12, 3X plus Y greater than equal to 8 and X and Y are also greater than equal to 0. Now first of all we consider the first constraint, we consider the line X plus 2Y equal to 10, we see that the 3 points lying on this line will be when X will be 0, Y will be 5 and X will be 10, Y will be 0, when X will be 2, Y will be 4. Now let us consider this line, this can be written as X plus Y equal to 6, 3 points lying on this line will be when X will be 0, Y will be 6, when X is 6, Y is 0, when X is 2, Y is 4. Now the third line is 3X plus Y is equal to 8, again we find out 3 points lying on this line, points will be when X is 0, Y is 8, when X is 8 by 3, Y is 0, when X is 1, Y is 5. Now using these points we plot these 3 lines and also considering the constraint that X is greater than equal to 0 and Y is greater than equal to 0, the required region will be in the first quadrant. Now this is how we plot all the points on the graph and we draw the lines. On the X axis we have taken 4X in kgs. On the Y axis we have taken 4Y again in kgs. Now this shaded region is the feasible region. So we can say that feasible solution is the shaded region A, G, H, F, whose N points are point A that is 10, 0, point G that is 2, 4, point H that is 1, 5 and point F that is 0, 10. So these are the points A, G, H and F. Now we see that Z is equal to 6X plus 10Y. Now at the point A that is 10, 0, Z will be equal to 6 into 10 plus 10 into 0 that is equal to 60. Now at the point G that is 2, 4, Z will be 6 into 2 plus 10 into 4 that is 12 plus 40 that is equal to 52. Now at point H that is 1, 5, Z will be 6 into 1 plus 10 into 5 that is equal to 56 and at the point F that is 0, 8 Z will be 6 into 0 plus 10 into 8 that is equal to 80. Thus the least cost is rupees 52 at the point G that is 2, 4, that is when 2 kg of food X and 4 kg of food Y are mixed together for desired diet then we will have the least cost. So our answer to this question is the least cost of the mixture which will produce the desired diet is rupees 52. Let us again see how we did this question. First we find out the linear programming problem in this case it is to minimize Z equal to 6X plus 10Y subject to these constraints. Now using these points we plot the three lines like this we have plotted three lines. Then we have found the feasible region this is this region because if we consider the constraint X is greater than equal to 0 and Y is greater than equal to 0 then the feasible region will be this shaded region. Now of these four points we find out that the cost will be minimum at the point G that is 2, 4 so we find out that when 2 kg of food X and 4 kg of food Y are mixed together for the desired diet then we get the least cost. So this is our answer to this question I hope that you understood the solution and enjoyed the session. Have a good day.