To find the minimum value for z=3x-(1/2)y, we need to find the point in the feasibility region where z is minimized.
First, let's graph the feasibility region:
The first constraint, x≥0, represents all points to the right of the y-axis.
The second constraint, y≤8, represents all points below or on the line y=8.
The third constraint, y≥x, represents all points above or on the line y=x.
The fourth constraint, y≥-1/2x+6, represents all points above or on the line y=-1/2x+6.
To find the feasibility region, we need to find the intersection points of the lines:
First, let's find the intersection point of y=8 and y=x:
8=x
So, the intersection point is (8, 8).
Next, let's find the intersection point of y=8 and y=-1/2x+6:
8=-1/2x+6
2x=2
x=1
Substituting x=1 into y=-1/2x+6:
y=-1/2(1)+6
y=5.5
So, the intersection point is (1, 5.5).
Finally, let's find the intersection point of y=-1/2x+6 and y=x:
-1/2x+6=x
-1x+12=2x
3x=12
x=4
Substituting x=4 into y=x:
y=4
So, the intersection point is (4, 4).
Now, let's graph the feasibility region:
The region is bounded by the x-axis, the line y=8, the line y=x, and the line y=-1/2x+6.
Next, let's evaluate z=3x-(1/2)y at each vertex of the feasibility region:
At (0, 8):
z=3(0)-(1/2)(8)
z=0-4
z=-4
At (1, 5.5):
z=3(1)-(1/2)(5.5)
z=3-2.75
z=0.25
At (4, 4):
z=3(4)-(1/2)(4)
z=12-2
z=10
The minimum value of z is -4 when x=0 and y=8.
Therefore, the correct answer is A. -4.
x≥0
y≤8
y≥x
y≥-1/2x+6
What is the minimum value for z=3x-1/2y over the feasibility region defined by the constraints shown above?
A. -4
B. -5
C. -3
D.- -6
1 answer