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?

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Let x represent the number of sacks of soybeans and y the number of sacks of oats.

The objective function is C=15x+5y (The total cost -- $15 per sack of soybeans and $5 per sack of oats.)

The constraints are :

`50x+15y>=120` or `10x+3y>=24`

`9x+5y>=27`

`5x+y>=10`

and the natural constraints `x,y>=0` .

Here is a graph of the feasible region:

The feasible region is open; it is above the red segments and contained in the 1st quadrant. We need only look at the junction points:

C(0,10)=50

C(1.2,4)=38

C(1.7,2.35)=37.25

C(3,0)=45

The actual point is `(39/23,54/23)` for a cost of 37.17.

The mathematical answer is `39/23` bags of soybeans and `54/23` bags of oats.

The real-life answer will be 2 bags each -- this is the closest lattice point ( a point with integer coordinates) within the feasible region.

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