To solve this optimization problem, we will define variables for the amount of each dog food brand to be used and set up inequalities based on the nutritional requirements, then solve to minimize the cost.

Step 1: Define the Variables

Let:

• $x$ = ounces of Brand A to be used per serving.
• $y$ = ounces of Brand B to be used per serving.

Step 2: Set Up the Constraints

We are given several constraints:

1. Total weight constraint: The total amount of food per serving should be no more than 8 ounces. $x + y \leq 8$

2. Nutrient I constraint: Each serving should contain at least 29 units of Nutrient I. $3x + 5y \geq 29$

3. Nutrient II constraint: Each serving should contain at least 20 units of Nutrient II. $4x + 2y \geq 20$

4. Non-negativity constraints: The amounts of Brand A and Brand B used must be non-negative. $x \geq 0 \quad \text{and} \quad y \geq 0$

Step 3: Set Up the Objective Function

The objective is to minimize the cost. The cost per ounce is:

• 4 cents for Brand A (so the cost contribution is $4x$).
• 5 cents for Brand B (so the cost contribution is $5y$).

Thus, the objective function is: $\text{Minimize } C = 4x + 5y$

Step 4: Solve the Linear Programming Problem

We have the following linear programming setup:

Objective:

Minimize $C = 4x + 5y$

Subject to:

x + y \leq 8 \\ 3x + 5y \geq 29 \\ 4x + 2y \geq 20 \\ x \geq 0 \\ y \geq 0 \end{cases}$$ Solving this problem involves graphing the feasible region defined by the constraints and finding the point(s) that minimize the objective function within this region. Let me calculate the feasible solution and find the optimal values of $$x$$ and $$y$$.The optimal solution for minimizing the cost while meeting the nutrient requirements is: - **Brand A**: 3 ounces - **Brand B**: 4 ounces - **Minimum Cost**: 32 cents per serving This combination meets all constraints and provides the necessary nutrients at the lowest cost. Would you like further details on any of these steps or an explanation on how each constraint is verified in this solution? Here are 5 follow-up questions that build on this problem: 1. How would the solution change if the cost of Brand B increased to 6 cents per ounce? 2. What if the total weight per serving limit increased from 8 ounces to 10 ounces? 3. How would a requirement increase for Nutrient I to 35 units per serving affect the cost? 4. Could other brands with different nutrient compositions provide a more cost-effective solution? 5. How can you verify that this solution is optimal within the feasible region? **Tip:** In optimization problems, always verify that the optimal solution meets all constraints by substituting values back into each constraint.