# An animal feed to be mixed from soybean meal and oats must contain at least 120lb of protein, 27lb of fat, and 10lb of mineral ash. Each sack of soybeans costs $15 and contains 50lb of protein,...

An animal feed to be mixed from soybean meal and oats must contain at least 120lb of protein, 27lb of fat, and 10lb of mineral ash. Each sack of soybeans costs $15 and contains 50lb of protein, 9lb of fat, and 5lb of mineral ash. Each sack of oats costs $5 and contains 15lb of protein, 5lb of fat and 1lb of mineral ash. How many sacks of each should be used to satisfy the minimum requirements at minimum cost? What is the minimum cost?

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### 1 Answer

Let x sack of soyabeans and y sack of oats are mixed in animal feed. Let C is cost of the feed.

Mathematical model of the problem is

Min C =15x+5y

s.t.

`50x+15y>=120` (Protein )

`9x+5y>=27` (fats)

`5x+y>=10` (mineral)

`x,y>=0`

Let plot the graph

Red : protein

Green : fats

blue : mineral

Green meets x-axis at P(3,0)

Red and green intersect at Q(1.7,2.33)

Red and Blue meet at R(1.2,4)

Blue meets y-axis at S(0,10)

C at P=15 x3+0=45

C at Q=15 x1.7+5x2.33=37.15

C at R= 15x1.2+5x4=38

c at S= 0+5x10=50

min( 45,37.15,38,50)=37.15 at Q (1.7,2.33)

Thus minimum cost is $ 37.15

and Soyabean =1.7 sacks

Oats= 2.33 sacks