question_answer 24)

A dietician wishes to mix together two kinds of foods X and Y in such a way that the mixture contains atleast 10 units of vitamin A, 12 units of vitamin B and 8 unis of vitamin C. The vitamin contents of one kg. food is given below :      

Food	Vitramin A	Vitamin B	Vitamin C
X Y	1 2	2 2	3 1

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

Answer:

Let x kg an y kg food of two kinds of foods X and Y to be mixed in a diet.       The contents of one kg. food of each kind as given below:

Food	Vitamin A	Vitamin B	Vitamin C	Cost
X	1	2	3	16
Y	2	2	1	20
Minimum Requirement	10	112	8

      The above L.P.P. is given as       Minimize, C = 16x + 20y, subject to the constraints x + 2y  2x + 2y  3x + y  8, y  0.       L1 : x + 2y = 10                    L2 : x + y = 6                            Here the cost is minimum at H(2, 4)             Since the region is unbounded therefore Rs.112 may be or may not be the minimum value of C.       For this draw graph of inequality            16x + 20y < 112       i.e. 4x + 5y < 28       L : 4x + 5y = 28       Clearly open half plane has no common point with the feasible region so minimum value of C is Rs.112.