Min P = 40x + 30y
10x + 16y ≥ 6,000
10x + 8y ≤ 3,600
x,y ≥ 0
Give the corners of the feasible set.

To find the corners (vertices) of the feasible set for the given linear programming problem, we need to determine the points of intersection between the constraints and check where these points satisfy all the inequalities.

### Step 1: Write down the constraints
1. \( 10x + 16y \geq 6000 \)
2. \( 10x + 8y \leq 3600 \)
3. \( x \geq 0 \)
4. \( y \geq 0 \)

### Step 2: Solve the inequalities as equations to find intersection points
We'll solve the following pairs of equations:

1. **Intersection of \( 10x + 16y = 6000 \) and \( 10x + 8y = 3600 \):**
   \[
   \begin{aligned}
   10x + 16y &= 6000 \quad \text{(Equation 1)} \\
   10x + 8y &= 3600 \quad \text{(Equation 2)}
   \end{aligned}
   \]

   Subtract Equation 2 from Equation 1:
   \[
   (10x + 16y) - (10x + 8y) = 6000 - 3600
   \]
   \[
   8y = 2400 \implies y = 300
   \]

   Substitute \( y = 300 \) into Equation 2:
   \[
   10x + 8(300) = 3600 \implies 10x + 2400 = 3600 \implies 10x = 1200 \implies x = 120
   \]
   So, the point of intersection is \( (120, 300) \).

2. **Intersection of \( 10x + 16y = 6000 \) and \( y = 0 \) (x-axis):**
   \[
   10x + 16(0) = 6000 \implies 10x = 6000 \implies x = 600
   \]
   So, the point is \( (600, 0) \).

3. **Intersection of \( 10x + 8y = 3600 \) and \( y = 0 \) (x-axis):**
   \[
   10x + 8(0) = 3600 \implies 10x = 3600 \implies x = 360
   \]
   So, the point is \( (360, 0) \).

4. **Intersection of \( x = 0 \) (y-axis) and \( 10x + 16y = 6000 \):**
   \[
   10(0) + 16y = 6000 \implies 16y = 6000 \implies y = 375
   \]
   So, the point is \( (0, 375) \).

5. **Intersection of \( x = 0 \) (y-axis) and \( 10x + 8y = 3600 \):**
   \[
   10(0) + 8y = 3600 \implies 8y = 3600 \implies y = 450
   \]
   So, the point is \( (0, 450) \).

### Step 3: Feasibility check
Now, let's identify the points that satisfy all the inequalities:

1. \( (120, 300) \)
2. \( (600, 0) \)
3. \( (360, 0) \)
4. \( (0, 375) \)

### Conclusion
The corners of the feasible set are:
- \( (120, 300) \)
- \( (600, 0) \)
- \( (360, 0) \)
- \( (0, 375) \)

Would you like more details, or do you have any questions about the process?

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**Related Questions:**
1. How do you find the optimal solution given the objective function and feasible set?
2. What role does the feasible region play in linear programming?
3. Can a linear programming problem have multiple optimal solutions?
4. How do boundary conditions affect the feasible set?
5. How would the feasible set change if one of the constraints is altered?

**Tip:** Always check all vertices of the feasible set to find the optimal solution, as it will always lie at one of these points.

This problem focuses on finding the vertices of the feasible region for a linear programming problem with constraints. The solution involves solving systems of linear equations, checking inequality conditions, and identifying the feasible set's corners.

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