Q32 of 97 Page 1376

A housewife wishes to mix together two kinds of food, X and Y, in such a way that the mixture contains at least 10 units of vitamin A, 12 units of vitamin B and 8 units of vitamin C.

The vitamin contents of 1 kg of each food are given below.



If 1 kg of food X cost ₹6 and 1 kg of food Y costs ₹10, find the minimum cost of the mixture which will produce the diet.



Let x and y be number of kilograms of food X and Y.


∴According to the question,


x + 2y , 2x + 2y


Minimize Z = 6x + 10y


The feasible region determined x + 2y , 2x + 2y is given by



The feasible region is unbounded. The corner points of feasible region are A(0,8) , B(1,5) , C(2,4) , D(10,0).The value of Z at corner points are



The minimum value of Z is 52 at point (2,4).


Hence, the diet should contain 2 kgs of food X and 4 kgs of food Y for the least cost of Rs. 52.


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A diet for a sick person must contain at least 4000 units of vitamins, 50 units of mineral and 1400 calories. Two food, A and B, are available at a cost of ₹4 and ₹3 per unit respectively. If one unit of A contains 200 units of vitamins, 1 unit of mineral and 40 calories, and 1 unit of B contains 100 units of vitamins, 2 units of mineral and 40 calories, find what combination of foods should be used to have the least cost.

 31

A diet for a sick person must contain at least 4000 units of vitamins, 50 units of mineral and 1400 calories. Two food, A and B, are available at a cost of ₹4 and ₹3 per unit respectively. If one unit of A contains 200 units of vitamins, 1 unit of mineral and 40 calories, and 1 unit of B contains 100 units of vitamins, 2 units of mineral and 40 calories, find what combination of foods should be used to have the least cost.

 32

A housewife wishes to mix together two kinds of food, X and Y, in such a way that the mixture contains at least 10 units of vitamin A, 12 units of vitamin B and 8 units of vitamin C.

The vitamin contents of 1 kg of each food are given below.



If 1 kg of food X cost ₹6 and 1 kg of food Y costs ₹10, find the minimum cost of the mixture which will produce the diet.


 33

A firm manufactures two types of product, A and B, and sells them at a profit of ₹5 per unit of type A and ₹3 per unit of type B. Each product is processed on two machines, M1 and M2 . one unit of type A requires one minute of processing time on M1 and two minutes of processing time on M2; whereas one unit of type B requires one minute of processing time on M1 and one minute on M2. Machines M1 and M2 are respectively available for at most 5 hours and 6 hours in a day. Find out how many units of each type of product the firm should produce a day in order to maximize the profit. Solve the problem graphically.