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CBSE Class 12 Math 2009 Solved Paper

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Question : 29
Total: 29
A diet is to contain at least 80 units of Vitamin A and 100 units of minerals. Two foods F1 and F2 are available. Food F1 cost Rs. 4 per unit and F2 costs Rs. 6 per unit. One unit of food F1 contains 3 units of Vitamin A and 4 units of minerals. One unit of food F2 contains 6 units of Vitamin A and 3 units of minerals. Formulate this as a linear programming problem and find graphically the minimum cost for diet that consists of mixture of these two foods and also meets the minerals nutritional requirements.
Solution:  
Let x be the number of units of food F1 and y be the number of units of food F2.
LPP is,
Minimize Z = 4x + 6y such that,
3x + 6y ≥ 80
4x + 3y ≥ 100
x , y ≥ 0
Representing the LPP graphically

Corner points are [0,
100
3
] , [24,
4
3
] , [
80
3
,0]
PointCost = 4x + 6y
(0,
100
3
)		 4 × 0 + 6 ×
100
3
 = 0 + 200 = 200
(24,
4
3
)		 4 × 24 + 6 ×
4
3
 = 96 + 8 = 104
(
80
3
,0)		 4 ×
80
3
 + 6 × 0 =
320
3
 + 0 = 106.67

From the table it is clear that, minimum cost is 104 and occurs at the point (24,
4
3
).
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