x>=0
y<=8
y>=x
y>=-1/2x+6
What is the minimum value for over the feasibility region defined by the constraints shown above?%0D%0A(1 point)%0D%0AResponses%0D%0A%0D%0A–4%0D%0A–4%0D%0A%0D%0A–5%0D%0A–5%0D%0A%0D%0A–3%0D%0A–3%0D%0A%0D%0A–6
To find the minimum value for y over the feasibility region, we need to find the minimum value for y in each of the given constraints.
First, we consider the constraint x >= 0. The minimum value for x in this constraint is 0. Therefore, y >= 0.
Next, we consider the constraint y <= 8. The maximum value for y in this constraint is 8. Therefore, y >= 8.
Next, we consider the constraint y >= x. Since x >= 0, the minimum value for y in this constraint is also 0. Therefore, y >= 0.
Finally, we consider the constraint y >= -1/2x + 6. Since x >= 0, the maximum value for y in this constraint is when x = 0. Plugging x = 0 into the equation, we get y >= 6.
Combining all the constraints, we find that the minimum value for y is 8.
Therefore, the minimum value for y over the feasibility region defined by the constraints is 8.
To find the minimum value for y over the feasibility region defined by the given constraints, we can follow these steps:
Step 1: Graph the feasibility region:
The given inequalities can be plotted on a coordinate plane to create the feasibility region.
The inequality x >= 0 represents all the points to the right of the y-axis (including the y-axis), which is the shaded region to the right of or on the y-axis.
The inequality y <= 8 represents all the points below or on the horizontal line y = 8, which is the shaded region below or on the line.
The inequality y >= x represents all the points above or on the line y = x, which is the shaded region above or on the line.
The inequality y >= -1/2x + 6 represents all the points above or on the line y = -1/2x + 6, which is the shaded region above or on the line.
By considering all the shaded regions, the feasibility region is the intersection of these regions. It is the part of the coordinate plane that satisfies all the given inequalities.
Step 2: Find the minimum value of y over the feasibility region:
To find the minimum value of y, we need to find the lowest point within the feasibility region.
Looking at the graph of the feasibility region, we can see that the point with the lowest y-coordinate is at the intersection of the lines y = x and y = -1/2x + 6.
Solving these two equations simultaneously gives us the x and y coordinates of the point of intersection.
x = -2, y = -2
Therefore, the minimum value for y over the feasibility region defined by the constraints is -2.
So, the correct answer is:
-2
To find the minimum value for y over the feasibility region defined by the constraints, we need to find the minimum value of y among the intersection points of the given inequalities.
The feasibility region is defined by the intersection of the following constraints:
1. x >= 0
2. y <= 8
3. y >= x
4. y >= -1/2x + 6
Let's find the intersection points of these constraints by solving the system of equations:
First, let's solve the equations for y = x and y = -1/2x + 6:
y = x ...(Equation 1)
y = -1/2x + 6 ...(Equation 2)
Equating Equation 1 and Equation 2:
x = -1/2x + 6
Multiplying both sides of the equation by 2:
2x = -x + 12
Combining like terms:
3x = 12
Dividing both sides by 3:
x = 12/3
x = 4
Substituting x = 4 into Equation 1 to find y:
y = x = 4
Therefore, the point (4, 4) is one of the intersection points.
Now, let's consider the other constraints:
Constraint 1: x >= 0
Since x must be greater than or equal to 0, the feasible values of x are x >= 0.
Constraint 2: y <= 8
Since y must be less than or equal to 8, the feasible values of y are y <= 8.
Constraint 3: y >= x
Since y must be greater than or equal to x, the feasible values of y are y >= x.
Constraint 4: y >= -1/2x + 6
This constraint is satisfied if y is greater than or equal to the line -1/2x + 6.
Now, let's plot the feasible region:
(4, 4)
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From the graph, we can see that the minimum value for y is 4, which occurs at the point (4, 4).
Therefore, the minimum value for y over the feasibility region is 4.