An animal feed to be mixed from soybean meal and oats must contain at least 168 lb of protein, 27 lb of fat, and 14 lb of mineral ash. each sack of soybeans costs $21 and contains 70 lb of protein, 9 lb of fat, and 7 lb of mineral ash. each sack of oats cost $7 and contains 21 lb of protein, 7lb of fat, and 1 lb of mineral ash. how many sacks of each should be used to satisfy the minimum requirements at minimum cos

[tex]\begin{array}{cccc}{} & {x} & {y} & {Total} & {Protein} & {70} & {21} & {168} &{Fats}& {9} & {7} & {27} & {Minerals} & {7} & {1} & {14}& {Cost} & {21} & {7} & {} \ \end{array}[/tex]

Where: x = Soybeans and y = Oats.

So, the system of equations are:

[tex]70x + 21y = 168[/tex]

[tex]9x +7y = 27[/tex]

[tex]7x + y = 14[/tex]

[tex]Cost = 21x + 7y[/tex]

The best way to solve this, is using graph

Plot the following equations on a graph, and get the points of intersection:

[tex]70x + 21y = 168[/tex]

[tex]9x +7y = 27[/tex]

[tex]7x + y = 14[/tex]

From the attached graph, we have:

[tex](x_1,y_1) = (1.636,2.545)[/tex]

[tex](x_2,y_2) = (1.775,1.575)[/tex]

[tex](x_3,y_3) = (2.023,1.256)[/tex]

Substitute each of the values of x's and y's in the cost function to get the minimum cost:

[tex]Cost = 21x + 7y[/tex]

[tex](x_1,y_1) = (1.636,2.545)[/tex]

[tex]Cost = 21 * 1.636 + 7 * 2.545[/tex]

[tex]Cost = 52.171[/tex]

[tex](x_2,y_2) = (1.775,1.575)[/tex]

[tex]Cost = 21 * 1.775 + 7 * 1.575[/tex]

[tex]Cost = 48.3[/tex]

[tex](x_3,y_3) = (2.023,1.256)[/tex]

[tex]Cost = 21 * 2.023 + 7 * 1.256[/tex]

[tex]Cost = 51.275[/tex]

The values of x and y that gives the minimum cost is:

[tex](x_2,y_2) = (1.775,1.575)[/tex]

and the minimum cost is:

[tex]Cost = 48.3[/tex]

Hence, the animal farm should buy 1.775 bags of soybeans and 1.575 bags of oats