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CBSE Class 12-science Answered

In order to ensure optimal health (and thus accurate test results), a lab technician needs to the dogs a daily diet containing a minimum of 24 grams (g) of fat, 36 g of carbohydrates, and 4 g of protein. But the dog should be fed no more than five kg of food a day. Rather than to order dog-food that is custom-blended, it is cheaper to order Food X and Food Y, and blend them for an optimal mix. Food X contains 8 g of fat, 12 g of carbohydrates, and 2 g of protein per kg, and costs Rs0.20 per kg. Food Y contains 12 g of fat, 12 g of carbohydrates, and 1 g of protein per kg, at a cost of Rs0.30 per kg. What is the optimal blend?
Asked by Topperlearning User | 18 Sep, 2014, 14:12: PM
answered-by-expert Expert Answer

Let us assume the following variables:

x: number of kg of Food X
y: number of kg of Food Y

Since negative amounts of either food exist, the first two constraints are: x > 0 and y > 0.

The other constraints are:

fat:        8x + 12y > 24
carbs:  12x + 12y > 36
protein:  2x +   1y >   4

Also, the maximum weight of the food is five kg, so

x + y < 5

The optimization equation will be the minimum value of the cost relation m C = 0.2x + 0.3y, b

After rearranging the inequalities, the system graphs as:

 

 

While testing the corners at (0, 4), (0, 5), (3, 0), (5, 0), and (1, 2), minimum cost of Rs 0.60 attains per daily serving, using three kg of Food X only.

Answered by | 18 Sep, 2014, 16:12: PM

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