A veterinarian has been asked to prepare a diet, x ounces of Brand A and y ounces of Brand B, 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 8 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 4 cents/ounce, and Brand B costs 7 cents/ounce. Determine how many ounces of each brand of dog food should be used per serving to meet the given requirements at a minimum cost.
(x, y) = 

    

What is the minimum cost (in cents per serving)? (Round your answer to the nearest cent.)
  cents per serving


To solve this optimization problem, we can set up a linear programming model to minimize the cost while meeting the nutrient requirements and weight restrictions.

### Step 1: Define Variables
Let:
- \( x \) = ounces of Brand A in the diet
- \( y \) = ounces of Brand B in the diet

### Step 2: Objective Function
The objective is to minimize the cost:
\[
\text{Minimize: } C = 4x + 7y
\]
where \( 4 \) cents per ounce for Brand A and \( 7 \) cents per ounce for Brand B.

### Step 3: Constraints
The problem provides constraints based on nutrient requirements and serving size:

1. **Total Serving Size Constraint:**
   \[
   x + y \leq 8
   \]

2. **Nutrient I Requirement:**
   Each ounce of Brand A provides 3 units, and each ounce of Brand B provides 5 units. We need at least 29 units:
   \[
   3x + 5y \geq 29
   \]

3. **Nutrient II Requirement:**
   Each ounce of Brand A provides 4 units, and each ounce of Brand B provides 2 units. We need at least 20 units:
   \[
   4x + 2y \geq 20
   \]

4. **Non-negativity Constraints:**
   \[
   x \geq 0, \quad y \geq 0
   \]

### Step 4: Solve the System of Constraints
We can solve this linear programming problem using the feasible region defined by the constraints and finding the point that minimizes the cost function \( C = 4x + 7y \).

Let’s calculate it step-by-step.

#### 1. Graph the Constraints
The feasible region is where all the constraints overlap. Points of intersection between these lines can be candidate solutions, as they are likely the vertices of the feasible region.

#### 2. Evaluate Objective Function at Vertices
We will check the values of \( C \) at each vertex to determine which one minimizes the cost.

### Solution Summary
Based on calculations from linear programming techniques (which can involve graphical or simplex methods), let’s calculate and determine:

- **Optimal values for \( x \) and \( y \)**
- **Minimum cost**

A veterinarian has been asked to prepare a diet, x ounces of Brand A and y ounces of Brand B, 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 8 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 4 cents/ounce, and Brand B costs 7 cents/ounce. Determine how many ounces of each brand of dog food should be used per serving to meet the given requirements at a minimum cost. (x, y) = ? What is the minimum cost (in cents per serving)? (Round your answer to the nearest cent.)

Determine the optimal amounts of Brand A and Brand B dog food to create a diet that meets nutrient requirements at minimal cost. This linear programming problem involves cost minimization, constraints on serving size and nutrient content, and uses inequalities to find the solution. Perfect for students learning optimization and linear programming techniques.

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