Math Problem Statement
what are the maximum value of z and what points corespond to the maximum value of z
Solution
To find the maximum value of and the corresponding points, we need to evaluate the objective function at the corner points of the feasible region determined by the constraints:
Constraints Recap:
Steps:
- Graph the constraints to find the feasible region.
- Determine the intersection points of the constraint lines, which are the corner points of the feasible region.
- Evaluate the objective function at each corner point.
Let's calculate the intersection points of the constraints analytically and then evaluate the objective function at each point to find the maximum value of .The feasible corner points and their corresponding values of the objective function are:
- Point with .
- Point with .
Maximum Value:
- The maximum value of is 430 at the point .
Would you like more details on the steps, or do you have any questions?
Related Questions:
- How would the optimal solution change if the constraints were modified slightly?
- What would happen to the feasible region if we added another constraint?
- How can we interpret this result in a real-world context?
- What if the objective function changed to ?
- How can we verify that the feasible region contains all relevant corner points?
Tip: Always recheck the constraints at the optimal point to ensure it satisfies all conditions.
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Math Problem Analysis
Mathematical Concepts
Linear Programming
Objective Functions
Feasible Region
Formulas
Objective function formula: z = 10x + 10y
Theorems
-
Suitable Grade Level
Advanced High School or College
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