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Question 1178973: Solve the linear programming problem.
Maximize P = 15x + 65y
Subject to x + 2y ≤ 16
x ≥ 0
y ≥ 0
What is the maximum value of P?
Please explain this homework question to me step-by-step
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! i solved graphically and i also solved using a simplex method tool.
the simplex method tool can be found at https://www.zweigmedia.com/RealWorld/simplex.html
the graphing tool can be found at https://www.desmos.com/calculator
the results from the simplex method tool are shown below.
the results from the graphing tool are shown below.
the simplex method tool is fairly straight forward and it give you the optimal solution only.
that optimal solution is at the (0,8) where the objective function of 15x + 65y is equal to 15 * 0 + 65 * 8 = 520.
the graph tool requires a little more work.
using the desmos.com calculator, you graph the opposite of the constraint inequalities.
the area of the graph that is not shaded is the region of feasibility.
the maximum / minimum value of the solution is at the corner points of the graph.
those corner points are (0,8), (16,0) and (0,0).
the objective function is evaluated at those corner points.
the objective function is p = 15x + 65y.
at (0,8), 15x + 65y = 15 * 0 + 65 * 8 = 520
at (16,0), 15x + 65y = 15 * 16 + 65 * 0 = 240
at (0,0), 15x + 65y = 15 * 0 + 65 * 0 = 0
both methods give you a maximum value of the objective function at (x,y) = 0,8).
the objective function is p = 5x + 65y
the contraint functions are:
x >= 0
y >= 0
x + 2y <= 16
you will be graphing the opposite of the constraint functions.
specifically, you would graph.
x <= 0
y <= 0
x + 2y >= 16
the reason for graphing the opposite of the inequalities is because, with the desmos.com software, the region of feasibility shows up clearly.
if you tried to graph the inequalities themselves, the feasible region would be much harder to spot.
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