A veterinarian has been asked to prepare a diet for a group of dogs to be used in a nutrition study at the School of Animal Science. It has been stipulated that each serving should be no larger than 10 oz and must contain at least 29 units of Nutrient I and 20 units of Nutrient II. The vet has decided that the diet may be prepared from two brands of dog food: Brand A and Brand B. Each ounce of Brand A contains 3 units of Nutrient I and 4 units of Nutrient II. Each ounce of Brand B contains 5 units of Nutrient I and 2 units of Nutrient II. Brand A costs 3 cents/oz and Brand B costs 4 cents/oz. Determine how many ounces of each brand of dog food should be used per serving to meet the given requirements at a minimum cost.

To determine the optimal number of ounces of Brand A and Brand B that should be used per serving, we need to set up a linear programming problem where we minimize the cost subject to the nutrient and serving size constraints.

Let:
- \( x \) = number of ounces of Brand A in each serving
- \( y \) = number of ounces of Brand B in each serving

### Objective Function
The cost of each ounce of Brand A is 3 cents, and each ounce of Brand B is 4 cents. Thus, the cost function to minimize is:
\[
\text{Minimize } C = 3x + 4y
\]

### Constraints
The problem provides the following requirements:

1. **Nutrient I requirement**: Each serving must contain at least 29 units of Nutrient I.
   - Brand A provides 3 units of Nutrient I per ounce.
   - Brand B provides 5 units of Nutrient I per ounce.
   - Thus, we have:
     \[
     3x + 5y \geq 29
     \]

2. **Nutrient II requirement**: Each serving must contain at least 20 units of Nutrient II.
   - Brand A provides 4 units of Nutrient II per ounce.
   - Brand B provides 2 units of Nutrient II per ounce.
   - So we have:
     \[
     4x + 2y \geq 20
     \]

3. **Serving size requirement**: Each serving should be no larger than 10 ounces.
   - Thus, we have:
     \[
     x + y \leq 10
     \]

4. **Non-negativity constraint**: The ounces of each brand cannot be negative.
   - Therefore:
     \[
     x \geq 0, \quad y \geq 0
     \]

### Summary of the Problem

The linear programming problem can be written as:

\[
\text{Minimize } C = 3x + 4y
\]

Subject to:
\[
\begin{cases}
3x + 5y \geq 29, \\
4x + 2y \geq 20, \\
x + y \leq 10, \\
x \geq 0, \\
y \geq 0.
\end{cases}
\]

### Solving the Problem

To solve this, we would typically use either the graphical method (if feasible) or the Simplex method to determine the values of \( x \) and \( y \) that minimize the cost function \( C \). Let's proceed by analyzing feasible points and checking the objective function at each vertex of the feasible region.

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Would you like a detailed explanation of the solution steps or a graphical solution to see the feasible region?

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**Related Questions:**

1. What is linear programming, and how is it applied in real-life scenarios like this?
2. How would changing the nutrient requirements affect the solution?
3. What if the cost of Brand A and Brand B were swapped? How would the optimal solution change?
4. How does the constraint on serving size impact the feasible solutions?
5. Can this problem be solved by another optimization method other than linear programming?

**Tip:** When setting up optimization problems, always define your variables and constraints clearly before forming the objective function.

Explore a linear programming solution to optimize a dog food diet for minimum cost. This problem involves constraints on nutrients and serving size, with solutions developed using cost minimization methods and feasible region analysis. Suitable for high school students in Grades 10-12.

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