Q3 of 31 Page 525

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

let x kg and y kg be amount of food X and Y respectively. Obviously x ≥ 0, y ≥ 0. Mathematical formulation of given problem is as follows:


Minimize Z = 16x + 20y ….(1)


Subject to the constraints,


x + 2y ≥ 10 …..(2)


x + y ≥ 6 …..(3)


3x + y ≥ 8 ….(4)


x ≥ 0, y ≥ 0 ….(5)


Now let us graph the feasible region of the system of inequalities (2) to (5). The feasible region (shaded) is shown in the fig. Here, we can observe that the feasible region is unbounded.



The coordinates of the corner points A(10,0),B(2,4), C(1,5)and D(0,8).



As, feasible region is unbounded so 112 may or may not be minimum value of Z.


So, 112 is the minimum value of Z, if the open half plane is determined by 16x + 20y <112 has no point in common with the feasible region. Otherwise z has no minimum value.


For this we will draw the graph of inequality 16x + 20y <112 or 4x + 5y<28. Here we find that 16x + 20y <112 has no point in common with the feasible region. Hence the minimum value of Z is 112 at B(2,4).


So, 2kg and 4kg be the amount of two foods P and Q respectively which are mixed to get minimum cost to Rs 112.


More from this chapter

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1

Refer to Example 9. How many packets of each food should be used to maximize the amount of vitamin A in the diet? What is the maximum amount of vitamin A in the diet?

 2

A farmer mixes two brands P and Q of cattle feed. Brand P, costing Rs 250 per bag, contains 3 units of nutritional element A, 2.5 units of element B and 2 units of element C. Brand Q costing Rs 200 per bag contains 1.5 units of nutritional element A, 11.25 units of element B, and 3 units of element C. The minimum requirements of nutrients A, B and C are 18 units, 45 units and 24 units respectively. Determine the number of bags of each brand which should be mixed in order to produce a mixture having a minimum cost per bag? What is the minimum cost of the mixture per bag?

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