- Asked 12/34
- 2074 Chaitra · 5 marks
- 2073 Shrawan · 5 marks
- 2071 Shrawan · 5 marks
Graphical method max:Z=5x₁+3x₂
Short answer
Plot constraints x1+2x2≤50, 2x1+x2≤40, x1,x2≥0. Corner points: (0,0), (20,0), (10,20), (0,25). Max Z=5x1+3x2 is 110 at (10,20).
Graphically maximize Z = 5x₁ + 3x₂ subject to the constraints x₁ + 2x₂ ≤ 50, 2x₁ + x₂ ≤ 40, x₁, x₂ ≥ 0.
Step 1: Draw the constraint lines. Replace each inequality by an equation, find the intercepts and decide which side of the line is allowed (test the origin).
| Constraint | Line | Meets x-axis | Meets y-axis | Allowed side |
|---|---|---|---|---|
| (1) x₁ + 2x₂ ≤ 50 | x₁ + 2x₂ = 50 | (50, 0) | (0, 25) | towards the origin |
| (2) 2x₁ + x₂ ≤ 40 | 2x₁ + x₂ = 40 | (20, 0) | (0, 40) | towards the origin |
Also x₁ ≥ 0 and x₂ ≥ 0, so the region lies in the first quadrant.
Step 2: Feasible region. The region satisfying all constraints is the shaded polygon below (dots), with corner points marked A, B, C, ….
29|
|
|D
|......
|.............
|...................C
|.....................
|.......................
|........................
|..........................
|............................
|.............................
|...............................
|.................................
|...................................
|....................................
0|A....................................B
+--------------------------------------------
0 23
Step 3: Corner points and the objective value. The optimum of a linear function over a convex polygon occurs at a corner point.
| Point | Coordinates | Lines meeting there | Z = 5x₁ + 3x₂ |
|---|---|---|---|
| A | (0, 0) | x₁ = 0, x₂ = 0 | 0 |
| B | (20, 0) | (2), x₂ = 0 | 100 |
| C | (10, 20) | (1), (2) | 110 |
| D | (0, 25) | (1), x₁ = 0 | 75 |
Step 4: Conclusion. The largest value of Z is 110, at C(10, 20).
Answer: Maximum Z = 110 at x₁ = 10, x₂ = 20.
✓ Answer checked by Claude Opus 5.5 (high effort)
Related questions
- Graphical method max:Z=7x₁+10x₂ · Mathematics III, chapter 5 · asked 12×
- Graphical method max:z=2x₁+3x₂ · Mathematics III, chapter 5 · asked 12×
- Graphical method max:Z=4x₁+5x₂ · Mathematics III, chapter 5 · asked 12×
- Graphical method max:Z=x₁+1.5x₂ · Mathematics III, chapter 5 · asked 12×
- Graphical max/min min:z=9x+40y · Mathematics III, chapter 5 · asked 12×
- Graphical max/min min:Z=5x₁+3x₂ · Mathematics III, chapter 5 · asked 12×
- Simplex method max:Z=3x₁+5x₂ · Mathematics III, chapter 5 · asked 17×
- Simplex method max:Z=3x₁+5x₂ · Mathematics III, chapter 5
- Big-M method max:P=2x₁+5x₂ · Mathematics III, chapter 5 · asked 15×
- Big-M method max:P=2x₁+5x₂ · Mathematics III, chapter 5 · asked 3×
- Big-M method max:P=2x₁+x₂+3x₃ · Mathematics III, chapter 5 · asked 17×
- Graphical max/min opt:Z=4x₁+5x₂ · Mathematics III, chapter 5
- EVA graphically; BCWS 9,50,000, BCWP 8,00,000 · Project Engineering, chapter 4
- Difference eq x(k+2)+2x(k+1)+3x(k)=0 · Applied Mathematics, chapter 2 · asked 5×
- Difference equation x(k+2)+3x(k+1)+2x(k)=0 · Digital Control System, chapter 2 · asked 5×