Q5 of 118 Page 30

A dietician mixes together two kinds of food in such a way that the mixture contains at least 6 units of vitamin A, 7 units of vitamin B, 11 units of vitamin C and 9 units of vitamin D. The vitamin contents of 1 kg of food X and 1 kg of food Y are given below :


One kg of food X costs ₹ 5, whereas one kg of food Y costs ₹ 8. Find the least cost of the mixture which will produce the desired diet.

The above information can be expressed with the help of the following table:



Let the quantity of foods X and Y be ‘x’ and ‘y’.


Cost of food X = 5x


Cost of food Y = 8y


Cost of the meal 5x+8y


Now,


⟹ x + 2y ≥ 6


i.e. the minimum requirement of Vitamin A in the foods X and Y is 6units, each of which has 1unit and 2 unit of Vitamin A.


⟹ x + y ≥ 7


i.e. the minimum requirement of Vitamin B in the two foods is 7units, each of which has 1 unit of Vitamin B.


⟹ x + 3y ≥ 11


i.e. the minimum requirement of vitamin C in the two foods is 11units, each of which has 1 unit and 3 units of vitamin C.


⟹ 2x + y ≥ 9


i.e. the minimum requirement of Vitamin D in the foods is 9units, each of which has 2 units and 1 unit of Vitamin D.


Hence, mathematical formulation of the LPP is as follows:


Find ‘x’ and ‘y’ that


Minimises Z = 5x + 8y


Subject to the following constraints:


(i) x + 2y ≥ 6


(ii) x + y ≥ 7


(iii) x + 3y ≥ 11


(iv) 2x + y ≥ 9


(v) x,y ≥ 0 (∵ quantity cant be negative)



The feasible region is unbounded.


The corner points of the feasible region are as follows:



Z is smallest at C(5,2)


Let us consider 5x + 8y ≤ 41.


As it has no intersection with the feasible region, the smallest value is the minimum value.


The minimum cost of the diet is ₹41


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