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This problem involves solving a linear programming (LP) problem, where the goal is to minimize the cost of making a dog treat using peanut butter and sweet potato while meeting nutritional requirements.

### Given Information:

- **Variables:**
  - Let \(x\) be the grams of peanut butter in the treat.
  - Let \(y\) be the grams of sweet potato in the treat.

- **Nutritional Requirements:**
  - At least 1.6 mg of vitamin E: 
    \[
    0.1x + 0.4y \geq 1.6
    \]
  - At least 30 calories: 
    \[
    5x + 2y \geq 30
    \]
  - At least 2 grams of protein: 
    \[
    0.2x + 0.2y \geq 2
    \]

- **Cost Function:**
  - Minimize the cost \(C\):
    \[
    C = 0.006x + 0.004y
    \]

### Step 1: Constraints
- The constraints are formed based on the given requirements:
  1. \(0.1x + 0.4y \geq 1.6\)  (Vitamin E requirement)
  2. \(5x + 2y \geq 30\)  (Calories requirement)
  3. \(0.2x + 0.2y \geq 2\)  (Protein requirement)
  4. Non-negative conditions: \(x \geq 0\), \(y \geq 0\)

### Step 2: Corner Points (Feasible Region)
From the problem, three corner points are already given:
- (16, 0)
- (0, 15)
- (8, 2)

The task also asks to find another corner point. To do this, you solve the system of equations formed by the constraints.

### Step 3: Solving for Minimum Cost
The next step is to evaluate the cost function at each corner point. The corner point that gives the lowest value for the cost function is the optimal solution.

Would you like me to calculate the corner points and determine the minimum cost from these?

Snoopy Services wishes to make healthy dog treats for their clients. Each dog treat should contain at least 1.6mg of vitamin E, at least 30 calories, and at least 2 grams of protein. The treats are primarily made of two ingredients: peanut butter and sweet potato. Each gram of peanut butter contains 0.1mg of vitamin E, 5 calories, 0.2 grams of protein, and costs 0.006 dollars. Each gram of sweet potato contains 0.4mg of vitamin E, 2 calories, 0.2 grams of protein, and costs 0.004 dollars. The goal is to minimize the cost while meeting the nutritional requirements.

This linear programming problem focuses on minimizing the cost of producing a dog treat using peanut butter and sweet potato, while satisfying nutritional constraints for vitamin E, calories, and protein. The problem involves setting up a cost function and solving for corner points.

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