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Question 1199234: Maximize P=3x + 2y
Subject to 2x +3y ≤ 12
x + y ≤ 8
x ≥ 0, y ≥ 0
Answer by math_tutor2020(3817)   (Show Source):
You can put this solution on YOUR website!
The shaded region for 2x+3y ≤ 12 is below the solid boundary 2x+3y = 12.
This boundary goes through (0,4) and (6,0) as the y and x intercepts respectively.
The shaded region for x+y ≤ 8 is below the solid boundary x+y = 8.
Since x ≥ 0 and y ≥ 0, we're focusing on the first quadrant in the northeast corner.
Here's what the shaded region looks like
This is the region that is below 2x+3y = 12 and x+y = 8.
Also, this region is to the right of the y axis, and above the x axis.
Points on the boundary are part of the shaded solution set.
Since all of 2x+3y = 12 is below x+y = 8, when working with quadrant Q1, this means we don't really need to worry about x+y ≤ 8
It's like saying "a building is less than 100 feet tall and also less than 30 feet tall" can be shortened to "a building is less than 30 feet tall".
The shaded feasibility region is a triangle with these corner points
A = (0, 0)
B = (0, 4)
C = (6, 0)
Each corner point is the result of intersecting two boundary line equations.
Ignore x+y = 8
For instance, point B is the intersection of the lines x = 0 and 2x+3y = 12.
Plug in x = 0 to get 2*0+3y = 12. That solves to y = 4.
So that's how to determine the coordinates for point B. Point C is similar but you'll plug in y = 0 and solve for x.
We check each of those corner points with the function P=3x + 2y
The goal of course is to see which gives the largest value of P.
Checking point A
plug in x = 0 and y = 0
P=3x + 2y
P=3*0 + 2*0
P=0
Checking point B
plug in x = 0 and y = 4
P=3x + 2y
P=3*0 + 2*4
P=8
Checking point C
plug in x = 6 and y = 0
P=3x + 2y
P=3*6 + 2*0
P=18
We reach the largest value of P at P = 18.
This occurs when x = 6 and y = 0.
Intuitively it makes sense because we're maxing out all resources to produce nothing but x, and none of y.
This is simply because the x coefficient is larger than the y coefficient.
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Answer:
Max value of P is 18
Occurs when x = 6 and y = 0
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