Math Problem Statement
Solution
To solve the linear programming problem in the uploaded image, let’s break it down step by step based on the problem's requirements:
Problem Overview:
Objective: Minimize
Subject to constraints:
- (non-negativity constraints)
Steps:
1. Find the x- and y-intercepts for each constraint:
-
For :
- Set : .
- Set : ().
-
For :
- Set : .
- Set : .
2. Graph the constraints:
- Plot the lines for and using their intercepts.
- Shade the feasible region that satisfies both constraints:
- Below (due to ).
- Above (due to ).
- Ensure the region is within the first quadrant (non-negativity).
3. Point of Intersection:
- Solve the equations and simultaneously:
- From , express : .
- Substitute into :
- Substitute into :
- Intersection point: .
4. Corner Points:
- Corner points of the feasible region:
- (intercept of ).
- (intersection of and ).
- (intercept of ).
5. Substitute into the Objective Function:
- Evaluate for each corner point:
- At : .
- At : .
- At : .
6. Formulate the Decision:
- Minimum occurs at with .
- Decision:
- ,
- ,
- .
Would you like a detailed graph or further explanation on any step?
Related Questions:
- How can you graph the feasible region manually?
- Why does the intersection point help solve the problem?
- How would the solution change if the objective was to maximize ?
- Can we use another method like the Simplex Method to solve this?
- What are practical scenarios where linear programming is used?
Tip:
Always confirm that your solution is in the feasible region by substituting back into all constraints.
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Math Problem Analysis
Mathematical Concepts
Linear Programming
Graphical Method
Optimization
Formulas
Objective Function: Z = 3x + 12y
Constraints: 5x + y ≤ 32, x + 3y ≥ 12, x, y ≥ 0
Theorems
Feasible Region
Linear Programming Optimization
Suitable Grade Level
College/Advanced High School
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