Question

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 content of one kg food is given below: Food Vitamin A Vitamin B Vitamin C X 1 2 3 Y 2 2 1 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?

Answer 1

Let the mixture contain x units of food X and the mixture contain y units of food Y.

The equation for vitamin A is given as,

x+2y≥10

The equation for vitamin B is given as,

2x+2y≥12 x+y≥6

The equation for vitamin C is given as,

3x+y≥8

We need to minimize the cost so we can use function which Minimize Z. Minimize Z=16x+20y (1)

All constraints are given as,

x+2y≥10 x+y≥6 3x+y≥8 x≥0,y≥0

x+2y≥10

x	0	10
y	5	0

x+y≥6

x	0	6
y	6	0

3x+y≥8

x	0	8 3
y	8	0

Plot the graph using equations of constraint.

Substitute the value of x and y in the equation (1) to check the max value,

Corner points	Value of Z
( 0,8 )	160
( 1,5 )	116
( 2,4 )	112(Minimum)
( 10,0 )	120

Where 112 may or may not be the minimum value of Z so we have to check graph inequality of 16x+20y<112.

There is no point common between feasible region and 16x+20y<112.

Thus, the cost will be minimum if the mixture contains 2 packets of food X and 4 packets of food Y and the minimum cost is Rs. 112.