An animal feed to be mixed from soybean meal and oats must contain at least $168 \mathrm{lb}$ of protein, $27 \mathrm{lb}$ of fat, and $14 \mathrm{lb}$ of mineral ash. Each sack of soybeans costs $\$ 21$ and contains $70 \mathrm{lb}$ of protein, $9 \mathrm{lb}$ of fat, and $7 \mathrm{lb}$ of mineral ash. Each sack of oats costs $\$ 7$ and contains $21 \mathrm{lb}$ of protein, $7 \mathrm{lb}$ of fat, and $1 \mathrm{lb}$ of mineral ash. How many sacks of each should be used to satisfy the minimum requirements at minimum cost? What is the minimum cost? (Round to the nearest cent as needed.)

Step 1 ：We denote the number of sacks of soybeans as x and the number of sacks of oats as y. The cost function to be minimized is \(21x + 7y\).

Step 2 ：We have the following constraints: \(70x + 21y \geq 168\) for protein, \(9x + 7y \geq 27\) for fat, and \(7x + y \geq 14\) for mineral ash.

Step 3 ：We also have the constraints \(x \geq 0\) and \(y \geq 0\) since we can't have a negative number of sacks.

Step 4 ：By solving this linear programming problem, we find that the optimal value of the cost function is approximately $51.28.

Step 5 ：This means that the minimum cost to satisfy the minimum requirements is $51.28.

Step 6 ：The values of x and y are approximately 2.02 and 1.26 respectively.

Step 7 ：This means that approximately 2 sacks of soybeans and 1 sack of oats should be used to satisfy the minimum requirements at minimum cost.

Step 8 ：Final Answer: The minimum cost to satisfy the minimum requirements is approximately \(\boxed{51.28}\).