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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)– 39
- 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 ≤...
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Minimise Z = 13x – 15y subject to the constraints : x + y ≤...
Problem Statement
We need to minimize the objective function Z = 13x + 15y subject to the following constraints:
1. x + y ≤ 7
2. 2x - 3y + 6 ≤ 0
3. x ≥ 0
4. y ≥ 0
Step 1: Graph the Constraints
- Constraint 1: x + y = 7
- Intercepts: (7, 0) and (0, 7)
- Constraint 2: 2x - 3y + 6 = 0
- Rearranging gives: y = (2/3)x + 2
- Intercepts: (0, -2) and (-3, 0) but we only consider non-negative values, so check where it intersects the first constraint.
Step 2: Find Intersection Points
- Set equations equal to find intersections:
- From x + y = 7 → y = 7 - x
- Substitute into 2x - 3(7 - x) + 6 = 0
- This simplifies to x = 3, y = 4 (intersection point)
- Check other boundary points formed by constraints.
- (0, 0), (0, 7), and (7, 0) are also potential vertices.
Step 3: Evaluate Z at Each Vertex
- (0, 0): Z = 13(0) + 15(0) = 0
- (0, 7): Z = 13(0) + 15(7) = 105
- (3, 4): Z = 13(3) + 15(4) = 39
- (7, 0): Z = 13(7) + 15(0) = 91
Step 4: Conclusion
The minimum value of Z occurs at the point (3, 4), yielding Z = 39. Thus, the correct answer is option 'C'.
We need to minimize the objective function Z = 13x + 15y subject to the following constraints:
1. x + y ≤ 7
2. 2x - 3y + 6 ≤ 0
3. x ≥ 0
4. y ≥ 0
Step 1: Graph the Constraints
- Constraint 1: x + y = 7
- Intercepts: (7, 0) and (0, 7)
- Constraint 2: 2x - 3y + 6 = 0
- Rearranging gives: y = (2/3)x + 2
- Intercepts: (0, -2) and (-3, 0) but we only consider non-negative values, so check where it intersects the first constraint.
Step 2: Find Intersection Points
- Set equations equal to find intersections:
- From x + y = 7 → y = 7 - x
- Substitute into 2x - 3(7 - x) + 6 = 0
- This simplifies to x = 3, y = 4 (intersection point)
- Check other boundary points formed by constraints.
- (0, 0), (0, 7), and (7, 0) are also potential vertices.
Step 3: Evaluate Z at Each Vertex
- (0, 0): Z = 13(0) + 15(0) = 0
- (0, 7): Z = 13(0) + 15(7) = 105
- (3, 4): Z = 13(3) + 15(4) = 39
- (7, 0): Z = 13(7) + 15(0) = 91
Step 4: Conclusion
The minimum value of Z occurs at the point (3, 4), yielding Z = 39. Thus, the correct answer is option 'C'.
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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)– 39d)– 34Correct answer is option 'C'. Can you explain this answer?
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