Questions: An animal feed to be mixed from soybean meal and oats must contain at least 120 lb of protein, 18 lb of fat, and 10 lb of mineral ash. Each sack of soybeans costs 15 and contains 50 lb of protein, 6 lb of fat, and 5 lb of mineral ash. Each sack of oats costs 5 and contains 15 lb of protein, 5 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? Are the answers reasonable? How could they be interpreted? The minimum cost is (Round to the nearest cent.)

Solution Steps

To solve this problem, we need to set up a system of linear inequalities based on the constraints given for protein, fat, and mineral ash. We will then use linear programming to minimize the cost function. The variables will represent the number of sacks of soybean meal and oats.

1. Define the variables:

   • Let \( x \) be the number of sacks of soybean meal.
   • Let \( y \) be the number of sacks of oats.

2. Set up the constraints:

   • Protein: \( 50x + 15y \geq 120 \)
   • Fat: \( 6x + 5y \geq 18 \)
   • Mineral ash: \( 5x + 1y \geq 10 \)

3. Define the cost function to minimize:

   • Cost: \( 15x + 5y \)

4. Use a linear programming solver to find the values of \( x \) and \( y \) that minimize the cost while satisfying the constraints.

Let \( x \) be the number of sacks of soybean meal and \( y \) be the number of sacks of oats. The constraints based on the nutritional requirements are:

1. Protein: \( 50x + 15y \geq 120 \)
2. Fat: \( 6x + 5y \geq 18 \)
3. Mineral ash: \( 5x + 1y \geq 10 \)

The cost function to minimize is given by:

\[ \text{Cost} = 15x + 5y \]

The solution to the linear programming problem yields the following results:

• Number of sacks of soybean meal: \( x \approx 2.0625 \)
• Number of sacks of oats: \( y \approx 1.1250 \)
• Minimum cost: \( \text{Cost} \approx 36.5625 \)

Final Answer