To graph the feasibility region, we need to plot the constraints and shade the region that satisfies all the constraints.
1. Graph the line 2x + y = 30:
- Find the x and y-intercepts by setting x = 0 and y = 0:
x = 0: 2(0) + y = 30, y = 30 → y-intercept: (0, 30)
y = 0: 2x + 0 = 30, x = 15 → x-intercept: (15, 0)
- Plot the points (0, 30) and (15, 0) and draw the line passing through them.
2. Graph the line x + 2y = 24:
- Find the x and y-intercepts by setting x = 0 and y = 0:
x = 0: 0 + 2y = 24, y = 12 → y-intercept: (0, 12)
y = 0: x + 2(0) = 24, x = 24 → x-intercept: (24, 0)
- Plot the points (0, 12) and (24, 0) and draw the line passing through them.
3. Shade the region that satisfies both constraints:
- To determine which side to shade, choose a test point not on the line.
- For example, use the origin (0, 0):
Substitute the coordinates (0, 0) into both inequalities:
2(0) + 0 ≤ 30 → 0 ≤ 30 (true)
0 + 2(0) ≤ 24 → 0 ≤ 24 (true)
- Since the origin satisfies both inequalities, shade the region below both lines.
Now we can identify the corner points of the feasibility region:
- (0, 0) is the intersection of both lines.
- (0, 12) is the intersection of the y-axis and the line x + 2y = 24.
- (15, 0) is the intersection of the x-axis and the line 2x + y = 30.
To find the point(s) that maximize the objective function z = 16x + 8y within the feasibility region, evaluate the objective function at each corner point:
- For (0, 0): z = 16(0) + 8(0) = 0
- For (0, 12): z = 16(0) + 8(12) = 96
- For (15, 0): z = 16(15) + 8(0) = 240
Thus, the point (15, 0) maximizes the objective function z = 16x + 8y within the feasibility region.