Q14 of 118 Page 31

A dietician 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 one kg food is given below :


One kg of food X costs ₹16 and one kg of food Y costs ₹ 20. Find the least cost of the mixture which will produce the required diet?

The above information can be expressed in the form of the following table:



Let the quantity of X and Y purchased be ‘x’ and ‘y’ kgs


Cost of X = 16x


Cost of Y = 20y


Cost of the mixture = 16x + 20y


Now,


⟹ x + 2y ≥ 10


i.e. the minimum requirement of Vitamin A from the mixture of X and Y is 10 units, each of which contains 1 unit and 2 units of Vitamin A respectively.


⟹ 2x + 2y ≥ 12


i.e. the minimum requirement of Vitamin B from the mixture of X and Y is 12 units, each of which contains 2units of vitamin B each.


⟹ 3x + y ≥ 8


i.e. the minimum requirement of vitamin C from the mixture of X and Y is 8 units, each of which contains 3 units and 1 unit of vitamin C respectively.


Hence, the mathematical formulation of the LPP is as follows :


Find ‘x’ and ‘y’ that:


Minimises Z = 16x + 20y


Subject to the following constraints:


(i) x + 2y ≥ 10


(ii) 2x + 2y ≥ 12


i.e. x + y ≥ 6


(iii) 3x + y ≥ 8


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



The feasible region is unbounded


The corner points of the feasible region is as follows:



Z is smallest at C(2,4)


Let us consider 16x + 20y ≤ 112


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


The minimum cost of the mixture is ₹112.


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