An animal feed to be mixed from soybean meal and oats must contain at least 168

Question

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 costs $7 and contains 21 lb of protein, 7 lb of fat, and 1 lb of mineral ash. How many sacks of each should be used to satisfy the minimum requirements at minimum cost?

What is the minimum cost?

How many sacks of each feed should be used to satisfy the minimum requirements at minimum cost?

How many sacks of soybean meal?

How many sacks of oats?

Explanation / Answer

ans=

Say there are x number of $21 sacks, and y number of $7 sacks. You then have 3 conditions to meet:

For protein: 70x + 21y => 168
For fat: 8x + 7y => 24
For ash: 7x + y => 14

If you plot these three conditions on a graph of x vs y, you find that each of these 3 conditions places limits of the number of sacks of feed you need. With increasing values of x, the limiting factor is in turn: ash, protein, fat. You can also find that all conditions are met with 3 sacks of cost $21 dollars and none of $7, or 14 sacks of cost $7 dollars and none of $21 - this immediately rules out the latter case on cost so it remains to consider the following 3 cases:

1 $21 dollar sack, from graph limiting factor is amount of ash => 7 $7 sacks = $70 cost
2 $21 dollar sacks, now limiting factor is protein => 2 $7 sacks, $56 total cost
3 $21 dollar sacks - no $7 sacks required, total cost $63

So you've now got minimum total cost and number of sacks required to meet this. Of course this problem has been made significantly easier by the fact (assumed) you can only take whole numbers of sacks of feed

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