Math Problem Statement
Solution
The problem shown is a linear programming (LP) problem, where you need to maximize the objective function subject to the given constraints:
Steps to solve:
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Graph the constraints:
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First, convert each inequality into an equation to determine the boundary lines.
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For , the intercepts are:
- , (solve for when )
- , (solve for when )
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For , the intercepts are:
- , (solve for when )
- , (solve for when )
-
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Find the feasible region:
- Plot the lines and shade the region that satisfies all constraints, including and .
- The feasible region will be a polygon where these lines intersect.
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Find the corner points of the feasible region:
- Solve the system of equations at the intersection points to determine the corner points of the polygon.
- The intersection of and can be solved simultaneously.
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Evaluate the objective function at each corner point:
- Calculate for each of the vertices of the feasible region.
- The maximum value of at these points will be the solution to the maximization problem.
I'll go ahead and calculate the corner points and the corresponding values of .
Solving for intersection points:
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Intersection of and :
Let's solve these equations using substitution or elimination.
From , divide by 2:
Now, subtract Equation 1 from :
Substitute into Equation 1:
So, the intersection point is .
Corner points:
The corner points of the feasible region are:
Evaluate at each corner point:
- At :
- At :
- At :
- At :
Conclusion:
The maximum value of is 21.5, which occurs at the point .
Would you like further details on this, or do you have any questions?
Here are 5 related questions:
- How do we handle linear programming problems with more than two variables?
- Can you explain the graphical method in more detail?
- What happens if the feasible region is unbounded?
- How do we identify if a problem has multiple optimal solutions?
- Can we use software tools to solve larger linear programming problems?
Tip: In linear programming, always check the boundary and intersection points of the constraints to maximize or minimize the objective function efficiently.
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Math Problem Analysis
Mathematical Concepts
Linear Programming
Graphical Method
Optimization
Formulas
p = 2x + 3y
3x + 8y ≤ 48
6x + 4y ≤ 42
Theorems
Feasible Region in Linear Programming
Corner Point Theorem
Suitable Grade Level
Grades 10-12, University Intro
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