Minimize z = 3x + 6y subject to:
4x + y ? 20
x + y ? 20
x + y ? 10
x ? 0
y ? 0
Graph the feasibility region.
Identify all applicable corner points of the feasibility region.
Find the point(s) (x,y) that minimizes the objective function z = 3x + 6y.
I assume those ? marks are < and > marks
corners
(0,0)
4x+y = 20
if x = 0, y = 20
if y = 0, x = 5
x+y = 20
if x = 0 , y = 20
if y = 0 , x = 20
this one is above and beyond the others when you graph it
x+y = 10
if x = 0 , y = 10
if y = 0, x = 10
intersection of
x+y = 10
4x+y = 20
---------
-3 x = -10
x = 10/3
y = 7/3
graph those and you will see that (0,0) is the minimum of 3x+6y
To find the feasibility region, we need to graph the given inequalities:
1. 4x + y ≥ 20
2. x + y ≥ 20
3. x + y ≤ 10
4. x ≥ 0
5. y ≥ 0
Let's start by graphing each inequality on a coordinate plane:
1. Graph the line 4x + y = 20 by finding its intercepts: (0, 20) and (5, 0). Since the inequality is "greater than or equal to," we shade the region above the line.
2. Graph the line x + y = 20 by finding its intercepts: (0, 20) and (20, 0). Shade the region above the line.
3. Graph the line x + y = 10 by finding its intercepts: (0, 10) and (10, 0). Shade the region below the line.
4. The x-axis already represents the condition x ≥ 0, so shade the entire region to the right of the y-axis.
5. Similarly, the y-axis already represents the condition y ≥ 0, so shade the entire region above the x-axis.
The overlapping region that satisfies all the conditions is called the feasibility region.
Now, to identify the corner points of the feasibility region, find the points where the boundaries intersect. These points are the vertices of the region.
Finally, to find the point(s) that minimize the objective function z = 3x + 6y, evaluate the objective function at each corner point and determine the minimum value.
Note: Without the specific values, it is not possible to provide the exact coordinates of the corner points or the point(s) that minimize the objective function. To find these values, you need to provide the specific values for the inequalities.