Math Problem Statement
what are the maximum points for the highest value of z
Solution
Let's calculate the maximum value of by first finding the corner points of the feasible region formed by the constraints. After that, we will evaluate the value of at each of these points.
Constraints Recap:
Step 1: Find the Intersection Points
We'll find the points of intersection for the pairs of constraint lines to determine the vertices of the feasible region.
1. Intersection of and :
- Solve these two equations simultaneously.
Multiply Equation 1 by 11 and Equation 2 by 6 to eliminate :
Add these equations:
Substitute back into Equation 1:
So, the intersection point is .
2. Intersection of and :
- Substitute into :
Since is not within the feasible region , this intersection does not contribute to a valid vertex in the feasible region.
3. Intersection of and :
- Substitute into :
Substitute into :
So, the intersection point is .
4. Intersection of and (substituting ):
- Substitute into :
So, the intersection point is .
5. Intersection of and :
- Substitute into :
So, the intersection point is .
Step 2: Evaluate at Each Vertex
The vertices of the feasible region are:
Now, evaluate at each vertex:
Conclusion:
The maximum value of occurs at the point .
Would you like any additional details or explanations?
Follow-up Questions:
- Would you like to see the feasible region graphically?
- Are you interested in verifying these calculations for the other constraints?
- Do you need further clarification on how the vertices were determined?
- Should we explore how changes in constraints affect the maximum value?
- Would you like to learn more about different methods for solving linear programming problems?
Tip:
In linear programming, the maximum or minimum value of the objective function always occurs at one of the vertices of the feasible region. Checking each vertex systematically ensures you find the optimal solution.
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Math Problem Analysis
Mathematical Concepts
Linear Programming
Feasible Region
Objective Function
Constraints
Formulas
Objective function formula: z = 10x + 10y
Theorems
-
Suitable Grade Level
Advanced High School
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