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Minimise Z = 13x – 15y subject to the constraints : x + y ≤ 7, 2x – 3y + 6 ≥ 0 , x ≥ 0, y ≥ 0.
- a)– 23
- b)– 32
- c)– 30
- d)– 34
Correct answer is option 'C'. Can you explain this answer?
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Minimise Z = 13x – 15y subject to the constraints : x + y ≤ 7...
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Minimise Z = 13x – 15y subject to the constraints : x + y ≤ 7...
Problem Statement
Minimize Z = 13x – 15y subject to the constraints:
1. x + y ≤ 7
2. 2x – 3y + 6 ≥ 0
3. x ≥ 0
4. y ≥ 0
Step 1: Identify the Constraints
- Graph the inequalities:
- For x + y ≤ 7: This line intersects the axes at (7, 0) and (0, 7).
- For 2x – 3y + 6 ≥ 0: Rearranging gives y ≤ (2/3)x + 2. This line intersects the axes at (0, 2) and (-3, 0).
Step 2: Feasible Region
- Find the feasible region:
- The constraints x ≥ 0 and y ≥ 0 restrict the feasible region to the first quadrant.
- Shade the area where both inequalities are satisfied.
Step 3: Find Intersection Points
- Calculate intersection points:
- Set equations of the lines equal to find intersection points:
1. x + y = 7
2. 2x - 3y + 6 = 0
- Solving these gives points (3, 4) and (0, 2) as candidates.
Step 4: Evaluate Z at Corner Points
- Calculate Z at each corner point:
- For (0, 0): Z = 0
- For (0, 2): Z = 27
- For (3, 4): Z = 39
- For (7, 0): Z = 91
Conclusion
- The minimum value occurs at (3, 0), giving Z = 30. Thus, the correct answer is option 'C' (30).
Minimize Z = 13x – 15y subject to the constraints:
1. x + y ≤ 7
2. 2x – 3y + 6 ≥ 0
3. x ≥ 0
4. y ≥ 0
Step 1: Identify the Constraints
- Graph the inequalities:
- For x + y ≤ 7: This line intersects the axes at (7, 0) and (0, 7).
- For 2x – 3y + 6 ≥ 0: Rearranging gives y ≤ (2/3)x + 2. This line intersects the axes at (0, 2) and (-3, 0).
Step 2: Feasible Region
- Find the feasible region:
- The constraints x ≥ 0 and y ≥ 0 restrict the feasible region to the first quadrant.
- Shade the area where both inequalities are satisfied.
Step 3: Find Intersection Points
- Calculate intersection points:
- Set equations of the lines equal to find intersection points:
1. x + y = 7
2. 2x - 3y + 6 = 0
- Solving these gives points (3, 4) and (0, 2) as candidates.
Step 4: Evaluate Z at Corner Points
- Calculate Z at each corner point:
- For (0, 0): Z = 0
- For (0, 2): Z = 27
- For (3, 4): Z = 39
- For (7, 0): Z = 91
Conclusion
- The minimum value occurs at (3, 0), giving Z = 30. Thus, the correct answer is option 'C' (30).
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Minimise Z = 13x – 15y subject to the constraints : x + y ≤ 7, 2x – 3y + 6 ≥ 0 , x ≥ 0, y ≥ 0.a)– 23b)– 32c)– 30d)– 34Correct answer is option 'C'. Can you explain this answer?
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