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
To find the minimum value for z = 3x - 1/2y over the given feasibility region, we need to consider the intersection points of the given constraints.
First, let's graph the constraints on a coordinate plane:
Constraint 1: x ≥ 0
This constraint represents the region to the right of the y-axis.
Constraint 2: y ≤ 8
This constraint represents the region below the line y = 8.
Constraint 3: y ≥ x
This constraint represents the region above the line y = x.
Constraint 4: y ≥ -1/2x + 6
This constraint represents the region above the line y = -1/2x + 6.
By analyzing the constraints, we can see that the feasible region is the overlapping region of all the shaded areas.
Since the objective function z = 3x - 1/2y is a linear function, its minimum value will occur at one of the boundary points of this region.
To find the intersection points, we can set up and solve the equations for the lines representing the constraints where they intersect:
Constraint 3 (y = x):
Constraint 4 (y = -1/2x + 6):
Equating the two equations: x = -1/2x + 6
2x = -x + 12
3x = 12
x = 4
Therefore, the intersection point is (4, 4).
Next, we need to check the values of z at the corner points of the feasible region.
Corner point 1: (0, 8)
z = 3(0) - 1/2(8) = -4
Corner point 2: (0, 6)
z = 3(0) - 1/2(6) = -3
Corner point 3: (4, 4)
z = 3(4) - 1/2(4) = 11
Since z = -4 is the smallest value among the corner points, the minimum value for z = 3x - 1/2y over the given feasibility region is -4.
Therefore, the correct answer is A. -4.
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.
To find the minimum value of z=3x-1/2y, we need to evaluate z at the corner points of the feasibility region formed by the given constraints.
Let's find the corner points by solving the system of equations formed by the four constraints:
1. x≥0
2. y≤8
3. y≥x
4. y≥-1/2x+6
To find the corner points, we need to identify the points where the feasible lines intersect.
First, we consider the intersection of the lines y=x and y=-1/2x+6.
Setting these two equations equal, we have:
x = -1/2x + 6
Simplifying the equation gives:
3/2x = 6
x = 6 * 2/3
x = 4
Substituting x=4 back into one of the equations, we can solve for y:
y = -1/2(4) + 6
y = -2 + 6
y = 4
So, the first corner point is (4, 4).
Next, we consider the intersection of the lines y=-1/2x+6 and y=8.
Setting these two equations equal, we have:
-1/2x + 6 = 8
-1/2x = 2
x = 2 * -2/1
x = -4
Substituting x=-4 back into one of the equations, we can solve for y:
y = -1/2(-4)+6
y = 2 + 6
y = 8
So, the second corner point is (-4, 8).
Now we can evaluate z = 3x - (1/2)y at the corner points:
For (4, 4):
z = 3(4) - (1/2)(4)
z = 12 - 2
z = 10
For (-4, 8):
z = 3(-4) - (1/2)(8)
z = -12 - 4
z = -16
Therefore, the minimum value of z=3x-(1/2)y over the feasibility region is -16, which corresponds to the point (-4, 8).
The correct answer is D.