An animal feed to be mixed from soybean meal and oats must contain at least 168 lb of​ protein, 24 lb of​ fat, and 14 lb of mineral ash. Each sack of soybeans costs $ 21 and contains 70 lb of​ protein, 8 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?

We are tasked with determining how many sacks of soybean meal and oats should be used to meet minimum nutritional requirements at the lowest cost.

Let:
- $ x $ be the number of sacks of soybean meal.
- $ y $ be the number of sacks of oats.

Each sack of soybeans contains:
- 70 lb of protein,
- 8 lb of fat,
- 7 lb of mineral ash.

Each sack of oats contains:
- 21 lb of protein,
- 7 lb of fat,
- 1 lb of mineral ash.

The total cost is:
- $21 per sack of soybean meal,
- $7 per sack of oats.

### Constraints:
We need at least:
- 168 lb of protein,
- 24 lb of fat,
- 14 lb of mineral ash.

### Objective Function (Minimize Cost):
\[
\text{Minimize:} \quad C = 21x + 7y
\]

### Protein Constraint:
\[
70x + 21y \geq 168
\]

### Fat Constraint:
\[
8x + 7y \geq 24
\]

### Mineral Ash Constraint:
\[
7x + 1y \geq 14
\]

We now solve this linear programming problem by graphing or using an optimization method (such as the Simplex method) to minimize the cost function under these constraints.

#### Step 1: Solve the inequalities.

1. **Protein constraint:** $ 70x + 21y \geq 168 $

   Divide through by 7 to simplify:
   \[
   10x + 3y \geq 24
   \]

2. **Fat constraint:** $ 8x + 7y \geq 24 $

3. **Mineral ash constraint:** $ 7x + y \geq 14 $

#### Step 2: Find the feasible region.

Solve these inequalities graphically or using algebraic methods to identify the points where the constraints intersect.

#### Step 3: Check vertices of the feasible region.

Once we have the feasible region from the system of inequalities, we will check the vertices to find the combination that minimizes the cost function $ C = 21x + 7y $.

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### 5 Related Questions:
1. How does linear programming apply to real-world optimization problems?
2. What methods can

An animal feed to be mixed from soybean meal and oats must contain at least 168 lb of​ protein, 24 lb of​ fat, and 14 lb of mineral ash. Each sack of soybeans costs $21 and contains 70 lb of​ protein, 8 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?

Solve an animal feed optimization problem using linear programming. Learn how to minimize the cost of mixing soybean meal and oats while meeting nutritional requirements for protein, fat, and mineral ash. The solution involves inequalities and a cost function. Suitable for Grades 11-12 or college-level students.

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