 Hello and welcome to the session. My name is Asha and I shall be helping you with the following question that says, A dealer wishes to purchase a number of fans and swing machines. He also has rupees 5,760 to invest and has space for at most 20 items. A fan cost him rupees 360 and a swing machine cost him rupees 240. His expectations is that he can sell a fan at a profit of rupees 22 and a swing machine at a profit of rupees 18. Assuming that he can sell all the items that he buys, how should he invest his money in order to maximize his profit, translate this problem mathematically and then solve it. Let's now start with the solution and first we shall formulate the LPP and then use the graphical method to solve it. We shall use fundamental extreme point theorem which states that the maximum or minimum value of a linear objective function over a convex polygon, that is feasible region determined by all the constraints, occur at some vertex of the polygon. For maximizing the profit of the dealer, we shall find the optimum number of fans and swing machine to be bought subject to the given constraints and this problem by using the fundamental extreme point theorem. Now, let the number of fans purchased is equal to x and the number of swing machines purchased is equal to y. Therefore, cost of x fans at the given rate of rupees 360 is equal to 360 into x and cost of y swing machines at the given rate of rupees 240 per machine is equal to 240 y. Therefore, cost of x fans and y swing machines is equal to 360 into x plus 240 into y. Now, the man only has rupees 5760 to invest. Therefore, the first constraint is 360 x plus 240 y is less than or equal to 5760. All it is further written as taking 120 common we have 3x plus 4y less than or equal to 120 into 48 or 3x plus 4y is less than or equal to 48. By canceling 120 which is the common factor from both the sides also x and y are greater than or equal to 0 since the number of fans and the number of swing machines purchased cannot be negative. Therefore, both x and y are greater than or equal to 0. Now, let us find the other equation we are given that the man has a space only to store 20 items. Therefore, total number of items cannot exceed 20 which implies that x plus y is less than or equal to 20. Therefore, the mathematical LPP is linear programming problem is maximize z as 22 into x plus 18 into y since the profit you can earn on one fan is rupees 22 and the profit on one swing machine is equal to rupees 18. Thus the total profit is equal to 22 into x plus 18 into y and we have to maximize it. Therefore, the objective function is maximize z is equal to 22x plus 18 y subject to the constraints as 2y less than or equal to 48 and x plus y is less than or equal to 20. Also, x is greater than or equal to 0 and y is also greater than or equal to 0. Now, let us draw the graph of first 3x plus 2y is equal to 48. Now, the points lying on it are make a small table xy, x is 0 then y is equal to 24 and if y is 0 then x is equal to 16. And now, let us find the points which lie on the line x plus y is equal to 20. So, here the points lying on this line are if x is 0 then y is 20 and if y is 0 then x is 20 thus the points lying on this line are 0 20 and 20 0. Now, let us draw the graph of both of these lines. First line is 3x plus 2y is equal to 48 and the first point which lie on this line is 0 24 that is when x is 0 y is 24. So, this point denotes 0 24 and the second point is 16 0. So, this point denotes 16 0. Now, let us join these two points to get the graph of line 3x plus 2y is equal to 48. This is 3x plus 2y is equal to 48. Now, let us draw the graph of x plus y is equal to 20. Now, the first point which lie on this line is 0 20 and second one is 20 0. Now, let us join both of these points 6 plus y is equal to 20. Now, after drawing the above two lines and considering the constraints we get the shaded portion in the graph as the feasible region which is a convex polygon with vertices 0, 20, 0, 0, 16, 0 and 8, 12 which is the point of intersection of these two lines. And according to the a fundamental extreme point theorem the maximum value of z will occur at any of these four points. Now let us substitute these four points to maximize z in the equation of z. Now z is equal to 22x plus 18y, so first let us take 0, 20, now z is equal to 22 into 0 plus 18 into 20 and this is equal to 360 at the point 0, 20. Now the second point of the vertex of the polygon is 0, 0, so here z is equal to 22 into 0 plus 18 into 0 and this is equal to 0. Now the third point is 16, 0 and the value of z at this point is 22 into 16 plus 18 into 0 and this is equal to 352 and at the last point which is 8, 12 the value of z is equal to 22 into 8 plus 18 into 12 and this gives 392. Now from these four values we find that 392 is the maximum value of z which is attained at the point 8, 12 therefore x is equal to 8 and y is equal to 12. Thus the man should purchase 8 fans and 12 swing machines to maximize his profit. Hence our answer is maximum profit is equal to rupees 392, the number of fans he should purchase is equal to 8 and swing machines is equal to 12. So this completes the session by intake care.