By graphing the system of constraints, find the values of x and y that minimize the objective function.
x+2y>=8
x>=2
y>=0
minimum for C = x + 3y
A. (8, 0)
B. (0, 10)
C. (10, 0)
To find the values of x and y that minimize the objective function, we need to graph the system of constraints and find the point that satisfies all the constraints and is closest to the origin.
First, let's graph the line x + 2y = 8. To do this, we can rewrite the equation in slope-intercept form as y = (-1/2)x + 4. Let's plot this line:
Next, let's graph the line x = 2. This is a vertical line passing through x = 2. Let's plot this line:
Finally, let's graph the line y = 0. This is a horizontal line passing through y = 0. Let's plot this line:
Now, let's find the region that satisfies all the constraints. This is the shaded region in the graph below:
To minimize the objective function C = x + 3y, we need to find the point in the shaded region that is closest to the origin. In other words, we need to find the point that minimizes the distance from the origin.
From the graph, it is clear that the point (8, 0) is the closest point to the origin that satisfies all the constraints. Therefore, the values of x and y that minimize the objective function are x = 8 and y = 0.
So, the answer is A. (8, 0).
To find the values of x and y that minimize the objective function, we need to graph the system of constraints and find the feasible region.
Let's plot the given constraints on a graph:
1) x + 2y >= 8:
To plot this constraint, we can rewrite it as y >= (8 - x)/2.
First, let's find the x-intercept and y-intercept:
At x = 0, y = 4 (x-intercept: (0, 4))
At y = 0, x = 8 (y-intercept: (8, 0))
Now, let's plot the constraint by drawing a line passing through these two intercepts and shading the region above the line.
2) x >= 2:
This constraint is a vertical line passing through x = 2. Shade the region to the right of this line.
3) y >= 0:
This constraint is a horizontal line passing through y = 0. Shade the region above this line.
Next, we need to find the feasible region, which is the overlapping region of all the shaded areas.
The feasible region is the area where all three shaded regions overlap. Now, we need to find the minimum point of the objective function within this feasible region.
The objective function is C = x + 3y.
To find the minimum value of C, we need to evaluate the objective function at the vertices of the feasible region.
The vertices of the feasible region are the points where the lines intersect:
A. (8, 0)
B. (0, 4)
C. (2, 0)
Now, let's evaluate the objective function at each vertex:
A: C = 8 + 3(0) = 8
B: C = 0 + 3(4) = 12
C: C = 2 + 3(0) = 2
From these calculations, we can see that the minimum value of C occurs at point C: (2, 0).
Therefore, the answer is C. (2, 0)
To find the values of x and y that minimize the objective function, we need to graph the system of constraints and then identify the feasible region. Once we have the feasible region, we can evaluate the objective function at each corner point to determine the minimum value.
First, let's graph the system of constraints:
1. Start by graphing the line x + 2y = 8:
- To graph this line, rewrite it in slope-intercept form (y = mx + b): 2y = -x + 8 -> y = (-1/2)x + 4
- Plot the intercept point (0, 4) on the y-axis.
- Determine another point by using the slope: for example, if x = 2, then y = 3. Plot this point.
- Draw the line passing through these two points.
2. Next, graph the line x = 2:
- This is a vertical line passing through x = 2. Draw the line.
3. Finally, graph the line y = 0:
- This is the x-axis. Draw the line.
Now, we have three lines on the coordinate plane that represent the given constraints. The feasible region, which satisfies all the constraints, is the intersection of all three regions on the graph.
In this case, the feasible region is the portion of the graph that is above the line x + 2y = 8, to the right of the line x = 2, and above the x-axis.
Next, we need to evaluate the objective function, C = x + 3y, at each of the corner points within the feasible region to find the minimum value.
The corner points of the feasible region are the points where the lines intersect. Let's identify these points and calculate the value of C = x + 3y at each one:
1. Point A: (8, 0)
C = 8 + 3(0) = 8
2. Point B: (0, 10)
C = 0 + 3(10) = 30
3. Point C: (10, 0)
C = 10 + 3(0) = 10
Comparing these values, we find that Point C, (10, 0), yields the minimum value for C = x + 3y.
Therefore, the answer is C. (10, 0).