Question 818226: Use graphical methods to solve the following linear programming problem:
Maximize z=2x+4y
subject to x+4y is less than or equal to 18
4x+2y is less than or equal to 16
X is greater than or equal to 0
Y is greater than or equal to 0
Found 2 solutions by Edwin McCravy, AnlytcPhil:
Answer by Edwin McCravy(20056)   (Show Source):
You can put this solution on YOUR website!
Maximize z=2x+4y
subject to x+4y ≦ 18
4x+2y ≦ 16
x ≧ 0
y ≧ 0
We form the equations of graphs of the boundary lines
lines by replacing the inequality symbols by equal
signs.
x+4y = 18 <---line thru points (0,4.5) and (6,3)
4x+2y = 16 <---line thru points (4,0) and (0,8)
x = 0 <---the y-axis
y = 0 <---the x-axis
We evaluate the objective function to maximize
at each corner point:
Corner z = 2x+4y
(0,0) z = 2(0)+4(0) = 0 <---minimum value
(0,4.5) z = 2(0)+4(4.5) = 18
(2,4) z = 2(2)+4(4) = 20 <---maximim value
(4,0) z = 2(4)+4(0) = 8
The maximum value of z is 20, and will occur at (2,4), when x=2 and y=4
Edwin
Answer by AnlytcPhil(1806)  (Show Source):
You can put this solution on YOUR website!
Maximize z=2x+4y
subject to x+4y ≦ 18
4x+2y ≦ 16
x ≧ 0
y ≧ 0
We form the equations of graphs of the boundary lines
lines by replacing the inequality symbols by equal
signs.
x+4y = 18 <---line thru points (0,4.5) and (6,3)
4x+2y = 16 <---line thru points (4,0) and (0,8)
x = 0 <---the y-axis
y = 0 <---the x-axis
We evaluate the objective function to maximize
at each corner point:
Corner z = 2x+4y
(0,0) z = 2(0)+4(0) = 0 <---- minimum value
(0,4.5) z = 2(0)+4(4.5) = 18
(2,4) z = 2(2)+4(4) = 20 <---- maximum value
(4,0) z = 2(4)+4(0) = 8
The maximum value of z is 20, and will occur at (2,4), when x=2 and y=4
Edwin
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