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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