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