Ruff, Inc. makes dog food out of chicken and grain. Chicken has 10 grams of protein and 5 grams of fat per ounce, and grain has 2 grams of protein and 2 grams of fat per ounce. A bag of dog food must contain at least 206 grams of protein and at least 166 grams of fat. If chicken costs 11¢ per ounce and grain costs 1¢ per ounce, how many ounces of each should Ruff use in each bag of dog food to minimize cost?

This is a linear programming problem where we want to minimize the cost of ingredients (chicken and grain) while satisfying certain constraints on protein and fat content.

### Step 1: Define the variables
Let:
- $ x $ = number of ounces of chicken used.
- $ y $ = number of ounces of grain used.

### Step 2: Formulate the objective function
The total cost of chicken and grain is:
\[
\text{Cost} = 11x + 1y
\]
We want to minimize this cost.

### Step 3: Set up the constraints
The dog food must meet the requirements for protein and fat:

1. **Protein constraint**: 
   - Chicken provides 10 grams of protein per ounce.
   - Grain provides 2 grams of protein per ounce.
   - The total protein must be at least 206 grams, so:
   \[
   10x + 2y \geq 206
   \]

2. **Fat constraint**:
   - Chicken provides 5 grams of fat per ounce.
   - Grain provides 2 grams of fat per ounce.
   - The total fat must be at least 166 grams, so:
   \[
   5x + 2y \geq 166
   \]

3. **Non-negativity constraints**: 
   Since negative amounts of chicken or grain do not make sense, we also have:
   \[
   x \geq 0 \quad \text{and} \quad y \geq 0
   \]

### Step 4: Solve the system of inequalities
We now need to solve this system using linear programming techniques (such as graphical methods, simplex, or software like Excel or Python's `scipy` library). Here's an outline of how to proceed:

1. **Graph the inequalities**: 
   Plot the constraints $ 10x + 2y = 206 $ and $ 5x + 2y = 166 $, and find the feasible region where both inequalities hold.

2. **Find the corner points**: 
   The feasible region's corner points can be calculated by solving the equations for the intersection points of the boundary lines.

3. **Evaluate the objective function at each corner**:
   Plug the corner points into the cost function $ 11x + 1y $ and find the minimum cost.

Let me calculate that for you.The solution to the problem is:

- Ruff, Inc. should use **0 ounces of chicken** and **103 ounces of grain** to minimize the cost.
- The minimum cost will be **$1.03** (103¢).

This combination meets the nutritional requirements at the lowest possible cost.

Would you like further details, or have any questions? Here are some related questions to explore:

1. What if the price of chicken decreases? How would it affect the solution?
2. How would the solution change if the protein or fat requirements increased?
3. Can we apply the simplex method manually to solve this?
4. What is the impact of introducing additional constraints, such as limiting the amount of grain?
5. What happens if both chicken and grain are equally priced?

**Tip:** Linear programming problems are often used in business to minimize costs or maximize profit under given constraints.

Learn how to solve a linear programming problem where Ruff, Inc. needs to minimize the cost of dog food ingredients while satisfying protein and fat content constraints. This problem uses linear inequalities and optimization techniques, suitable for Grades 10-12 students.

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