13. CONSIDER THE FOLLOWING LINEAR PROGRAM:
Minimize C = 75P + 32Q
Subject To:
9P + 5Q > = 45
0 <= 6
P <=3
(1)
(2)
(3)
P> =0, Q> =0
If you place P in the horizontal axis, which of the following is NOT a vertex of the set of feasible solutions? 

To find which vertex is **NOT** part of the feasible solution, we need to go through the constraints and find the vertices of the feasible region.

### The Linear Program is:
Objective Function:
\[
C = 75P + 32Q
\]

Subject to the constraints:
1. \( 9P + 5Q \geq 45 \)  (Inequality)
2. \( 0 \leq P \leq 3 \)  (Bounds on \(P\))
3. \( P \geq 0, Q \geq 0 \)  (Non-negativity constraints)

#### Steps:

### Step 1: Graphical Representation of the Constraints
1. **Constraint 1: \(9P + 5Q \geq 45\)**

   - Rewrite this in slope-intercept form to find the boundary line:
     \[
     Q = \frac{45 - 9P}{5}
     \]
     This is a line with a negative slope of \(-\frac{9}{5}\) and a y-intercept of \(9\) (where \(P = 0\)).

2. **Constraint 2: \(0 \leq P \leq 3\)**
   
   This restricts \(P\) to be between 0 and 3.

3. **Non-negativity constraints \(P \geq 0\) and \(Q \geq 0\)**

   These limit the feasible region to the first quadrant.

### Step 2: Identify Intersection Points (Vertices)
The vertices are the intersection points of the boundary lines. The primary candidates are:

- **Intersection of \( 9P + 5Q = 45 \) with \( P = 0 \):**

  Plug \(P = 0\) into the equation \(9P + 5Q = 45\):
  \[
  5Q = 45 \quad \Rightarrow \quad Q = 9
  \]
  Vertex: \((0, 9)\)

- **Intersection of \( 9P + 5Q = 45 \) with \( P = 3 \):**

  Plug \(P = 3\) into \(9P + 5Q = 45\):
  \[
  9(3) + 5Q = 45 \quad \Rightarrow \quad 27 + 5Q = 45 \quad \Rightarrow \quad 5Q = 18 \quad \Rightarrow \quad Q = \frac{18}{5} = 3.6
  \]
  Vertex: \((3, 3.6)\)

- **Intersection of \( P = 3 \) and \( Q = 0 \):**

  This gives the vertex: \((3, 0)\).

- **Intersection of \( P = 0 \) and \( Q = 0 \):**

  This is the origin: \((0, 0)\).

### Step 3: List of Vertices
The feasible vertices are:
1. \( (0, 9) \)
2. \( (3, 3.6) \)
3. \( (3, 0) \)
4. \( (0, 0) \)

### Conclusion: Which is **NOT** a Vertex?
Now, based on the calculated vertices, any point outside this set is not a vertex. Compare with the given options (if provided), and find the one that is not part of this list.

Would you like to check specific points or ask for more details?

### Related Questions:
1. How do we determine if a solution is optimal in linear programming?
2. How can we handle inequalities that involve both greater-than and less-than constraints?
3. What are the graphical methods used to solve linear programming problems?
4. How do slack variables help in converting inequalities to equations?
5. How does the Simplex Method compare to graphical solutions?

**Tip:** When solving a linear programming problem graphically, always ensure that you properly account for all the constraints, including bounds and non-negativity conditions.

13. CONSIDER THE FOLLOWING LINEAR PROGRAM: Minimize C = 75P + 32Q Subject To: 9P + 5Q >= 45 0 <= 6 P <= 3 P >= 0, Q >= 0 If you place P in the horizontal axis, which of the following is NOT a vertex of the set of feasible solutions?

This problem covers a linear programming scenario where the objective is to minimize C = 75P + 32Q, subject to constraints like 9P + 5Q >= 45. Learn how to graph the constraints, find feasible vertices, and determine which point is NOT a vertex. Suitable for undergraduate students in operations research or linear programming courses.

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