By graphing the system of constraints, find the values of x and y that maximize the objective function.
2x+y<=300
x+y<=200
x=>0
y=>0
maximum for p=x+2y
Graph the first two as lines
y= -2x+300 and
y=-x+300
then graph the positive x axis, and the positive y axis. Now the region within that four sided figure is the allowed region for x and y. However, we have a nice theorem which states that maximum and minimum of any objective funtion will occur on the boundry points. I can tell you the minimium will be at (0,0). The maximum with be at one of the other corners. You can see on the graph the four other corners, or you can solve for those points. Which one makes x+2y the greatest
To graph the system of constraints, we can start by graphing each individual equation on a coordinate plane.
The first constraint, 2x + y ≤ 300, can be rewritten as y ≤ -2x + 300. To plot this on a graph, we can draw a line with a slope of -2 that passes through the point (0, 300).
The second constraint, x + y ≤ 200, can be plotted as a line with a slope of -1 that passes through the point (0, 200).
Next, we need to identify the feasible region, which represents the intersection of the shaded regions formed by both lines. This region will satisfy all the given constraints.
To find the values of x and y that maximize the objective function p = x + 2y, we need to locate the highest point within the feasible region. The objective function represents the value we are trying to maximize.
Since the objective function is a linear equation with a positive slope, the highest point will be on the edge of the feasible region where the slope is steepest. In this case, it will be on the line where 2y = x.
To find this point, we can set up a system of equations by substituting 2y from the equation 2y = x into the other constraint equation x + y ≤ 200:
x + 2y = 200 (substituting 2y for x)
2y + y = 200
3y = 200
y = 66.67
Substituting this value back into the equation 2y = x, we can find the value of x:
2(66.67) = x
x = 133.33
So, the values of x and y that maximize the objective function p = x + 2y are approximately x = 133.33 and y = 66.67.