Math Problem Statement
Solution
This is a linear programming problem where you need to maximize subject to the following constraints:
Steps to Solve:
-
Graph the Constraints:
- : This line intersects the -axis at and the -axis at .
- : This line intersects the -axis at and the -axis at .
- : Restrict the feasible region to the first quadrant.
-
Find Corner Points: The feasible region is bounded by the intersection of these lines and the - and -axes:
- Solve the system of equations:
- and to find the intersection point.
- Intersections with axes and each other form the vertices of the feasible region.
- Solve the system of equations:
-
Evaluate at Each Vertex: Substitute the coordinates of each corner point into .
Let me calculate the exact corner points and the optimal value. sympy import symbols, Eq, solve
Define variables
x, y = symbols('x y', nonnegative=True)
Define constraints as equations
eq1 = Eq(x + 6y, 20) # x + 6y = 20 eq2 = Eq(6x + y, 15) # 6x + y = 15
Intersection of the two lines
intersection = solve((eq1, eq2), (x, y))
Intersection points with axes
x_intercept_eq1 = solve(eq1.subs(y, 0), x) # x-intercept of eq1 y_intercept_eq1 = solve(eq1.subs(x, 0), y) # y-intercept of eq1 x_intercept_eq2 = solve(eq2.subs(y, 0), x) # x-intercept of eq2 y_intercept_eq2 = solve(eq2.subs(x, 0), y) # y-intercept of eq2
Collect vertices of feasible region
vertices = [ (x_intercept_eq1[0], 0), # x-intercept of eq1 (0, y_intercept_eq1[0]), # y-intercept of eq1 (x_intercept_eq2[0], 0), # x-intercept of eq2 (0, y_intercept_eq2[0]), # y-intercept of eq2 (intersection[x], intersection[y]) # intersection of eq1 and eq2 ]
Evaluate p = x + 2y at each vertex
p_values = [(v[0], v[1], v[0] + 2*v[1]) for v in vertices]
vertices, p_valuesThe vertices of the feasible region and the corresponding values of are as follows:
- :
- :
- :
- : (maximum)
- :
Optimal Solution:
- The maximum value of is 30, occurring at .
Would you like a graph of this feasible region or more explanation?
Related Questions:
- How are the corner points of the feasible region calculated geometrically?
- Can this problem be solved using matrix methods (simplex method)?
- What happens if we change the objective function to ?
- How does the feasibility of the solution change if one constraint becomes an equality?
- How do you interpret the solution in real-world optimization problems?
Tip:
Always check all vertices of the feasible region when solving linear programming problems, as the optimum is guaranteed to occur at one of them.
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Math Problem Analysis
Mathematical Concepts
Linear Programming
Graphing Inequalities
Optimization
Formulas
Objective Function p = x + 2y
Constraints x + 6y ≤ 20
Constraints 6x + y ≤ 15
Theorems
Fundamental Theorem of Linear Programming
Corner Point Theorem
Suitable Grade Level
Grades 11-12
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