MaxZ=40x+60ySujeto a:3x+2y≤2000x+2y≤1000x,y≥0

We are given the linear programming problem:

\[
\text{Maximize } Z = 40x + 60y
\]
subject to the constraints:
1. \( 3x + 2y \leq 2000 \)
2. \( x + 2y \leq 1000 \)
3. \( x, y \geq 0 \)

### Step 1: Graphical Method (Identify Feasible Region)

First, we need to graph the constraints to find the feasible region.

1. \( 3x + 2y \leq 2000 \)
   - If \( x = 0 \), \( 2y = 2000 \Rightarrow y = 1000 \)
   - If \( y = 0 \), \( 3x = 2000 \Rightarrow x \approx 666.67 \)

2. \( x + 2y \leq 1000 \)
   - If \( x = 0 \), \( 2y = 1000 \Rightarrow y = 500 \)
   - If \( y = 0 \), \( x = 1000 \)

Now plot these lines and find the intersection points of the feasible region.

### Step 2: Find Intersection Points

We solve the system of equations for the intersections of the lines:
1. \( 3x + 2y = 2000 \)
2. \( x + 2y = 1000 \)

Subtract the second equation from the first:
\[
(3x + 2y) - (x + 2y) = 2000 - 1000
\]
\[
2x = 1000 \Rightarrow x = 500
\]

Substitute \( x = 500 \) into \( x + 2y = 1000 \):
\[
500 + 2y = 1000 \Rightarrow 2y = 500 \Rightarrow y = 250
\]

So, the intersection point is \( (500, 250) \).

### Step 3: Evaluate Objective Function at Corner Points

The corner points of the feasible region are:
1. \( (0, 0) \)
2. \( (0, 500) \)
3. \( (666.67, 0) \)
4. \( (500, 250) \)

Now, we evaluate \( Z = 40x + 60y \) at each corner point:

1. At \( (0, 0) \): \( Z = 40(0) + 60(0) = 0 \)
2. At \( (0, 500) \): \( Z = 40(0) + 60(500) = 30,000 \)
3. At \( (666.67, 0) \): \( Z = 40(666.67) + 60(0) = 26,666.8 \)
4. At \( (500, 250) \): \( Z = 40(500) + 60(250) = 20,000 + 15,000 = 35,000 \)

### Step 4: Conclusion

The maximum value of \( Z \) occurs at \( (500, 250) \), and the maximum value of \( Z \) is:

\[
Z_{\text{max}} = 35,000
\]

---

Do you want more details on any step? Here are 5 related questions you might consider:

1. How do we handle linear programming problems with more than two variables?
2. Can you explain how to derive the corner points graphically?
3. What methods can be used if we cannot solve linear programming graphically?
4. How would the solution change if the objective function had a different form?
5. How do we confirm that the solution found is optimal?

**Tip:** In linear programming, always check the feasibility of each point before concluding it's a solution.

MaxZ=40x+60y Sujeto a: 3x+2y≤2000, x+2y≤1000, x,y≥0

Solve the linear programming problem by maximizing Z = 40x + 60y subject to constraints using the graphical method. This step-by-step solution covers how to find the feasible region, calculate intersection points, and evaluate the objective function at corner points. Suitable for Grades 11-12 students.

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