Solve the problem graphically
Maximize `8000 x + 7000 y`
Subject to `3x + y le 66`
`x + y le 45`
`x le 20`
`y le 40`
and `x, y ge 0`

Answer

To solve the given linear programming problem graphically, we will follow these steps:

### Step 1: Identify the Objective Function and Constraints
We need to maximize the objective function:
\[ Z = 8000x + 7000y \]
Subject to the constraints:
1. \( 3x + y \leq 66 \)
2. \( x + y \leq 45 \)
3. \( x \leq 20 \)
4. \( y \leq 40 \)
5. \( x, y \geq 0 \)

### Step 2: Graph the Constraints
We will graph each constraint on the coordinate plane.

1. **For \( 3x + y = 66 \)**:
- When \( x = 0 \), \( y = 66 \) (Point: (0, 66))
- When \( y = 0 \), \( x = 22 \) (Point: (22, 0))
- The line will be drawn between these points, and the region below this line is the feasible region for this constraint.

2. **For \( x + y = 45 \)**:
- When \( x = 0 \), \( y = 45 \) (Point: (0, 45))
- When \( y = 0 \), \( x = 45 \) (Point: (45, 0))
- The line will be drawn between these points, and the region below this line is the feasible region for this constraint.

3. **For \( x = 20 \)**:
- This is a vertical line at \( x = 20 \).

4. **For \( y = 40 \)**:
- This is a horizontal line at \( y = 40 \).

5. **For \( x \geq 0 \) and \( y \geq 0 \)**:
- This means we are only considering the first quadrant of the graph.

### Step 3: Determine the Feasible Region
The feasible region is the area where all the constraints overlap. It is bounded by the lines we have drawn and is located in the first quadrant.

### Step 4: Identify the Corner Points of the Feasible Region
The corner points of the feasible region can be found by solving the equations of the lines that intersect at those points.

1. **Intersection of \( 3x + y = 66 \) and \( x + y = 45 \)**:
\[ 3x + y = 66 \quad (1) \]
\[ x + y = 45 \quad (2) \]
Subtract (2) from (1):
\[ 2x = 21 \implies x = 10.5 \]
Substitute \( x \) back into (2):
\[ 10.5 + y = 45 \implies y = 34.5 \]
So, one corner point is \( (10.5, 34.5) \).

2. **Intersection of \( 3x + y = 66 \) and \( y = 40 \)**:
\[ 3x + 40 = 66 \implies 3x = 26 \implies x = \frac{26}{3} \approx 8.67 \]
So, another corner point is \( \left(\frac{26}{3}, 40\right) \).

3. **Intersection of \( x + y = 45 \) and \( y = 40 \)**:
\[ x + 40 = 45 \implies x = 5 \]
So, another corner point is \( (5, 40) \).

4. **Intersection of \( x + y = 45 \) and \( x = 20 \)**:
\[ 20 + y = 45 \implies y = 25 \]
So, another corner point is \( (20, 25) \).

5. **Intersection of \( x = 20 \) and \( y = 40 \)**:
This point is \( (20, 40) \).

### Step 5: Evaluate the Objective Function at Each Corner Point
Now we will evaluate \( Z = 8000x + 7000y \) at each corner point:

1. At \( (0, 0) \):
\[ Z = 8000(0) + 7000(0) = 0 \]

2. At \( (20, 0) \):
\[ Z = 8000(20) + 7000(0) = 160000 \]

3. At \( (10.5, 34.5) \):
\[ Z = 8000(10.5) + 7000(34.5) = 84000 + 241500 = 325500 \]

4. At \( (5, 40) \):
\[ Z = 8000(5) + 7000(40) = 40000 + 280000 = 320000 \]

5. At \( (20, 25) \):
\[ Z = 8000(20) + 7000(25) = 160000 + 175000 = 335000 \]

6. At \( (20, 40) \):
\[ Z = 8000(20) + 7000(40) = 160000 + 280000 = 440000 \]

### Step 6: Determine the Maximum Value
The maximum value occurs at the point \( (20, 40) \) with:
\[ Z = 440000 \]

### Final Answer
The maximum value of the objective function is \( 440000 \) at the point \( (20, 40) \).

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