A breed of cattle needs at least 10 proteins and 8 fats units per day. Feed type I provides 6 protein and 2 fat units at $4 per bag. Feed II provides 2 protein and 3 fat units at $3 per bag. Find the minimum cost.

Let the number of bags of feed type I to be used be x and the number of bags of feed type II to be used be y, then:

We are to minimize:

C = 4x + 3y

subject to the following constraints:

[tex]6x+2y\geq10 \\ \\ 2x+3y\geq8 \\ \\ x\geq0,\ y\geq0[/tex]

From the graph of the 4 constraints above, the corner points are (0, 5), (1, 2), (4, 0).

Testing the objective function for the minimum corner point we have:

For (0, 5):
C = 4(0) + 3(5) = $15

For (1, 2):
C = 4(1) + 3(2) = 4 + 6 = $10

For (4, 0):
C = 4(4) + 3(0) = $16.

Therefore, the combination that yields the minimum cost is 1 bag of type I feed and 2 bags of type II feed.