Maximize z = 16x + 8y subject to:
2x + y ≤ 30
x + 2y ≤ 24
x ≥ 0
y ≥ 0
Graph the feasibility region.
Identify all applicable corner points of the feasibility region.
Find the point(s) (x,y) that maximizes the objective function z = 16x + 8y.
To graph the feasibility region, we need to plot the lines representing the inequalities and shade the region that satisfies all the given conditions.
First, let's plot the lines 2x + y = 30 and x + 2y = 24 on a graph. To do this, we can find their respective x-intercepts and y-intercepts.
For 2x + y = 30:
Setting y = 0, we get 2x + 0 = 30, which gives x = 15. So one point is (15, 0).
Setting x = 0, we get 0 + y = 30, which gives y = 30. So another point is (0, 30).
Plotting these points and drawing a line through them, we have one boundary line.
For x + 2y = 24:
Setting y = 0, we get x + 0 = 24, which gives x = 24. So one point is (24, 0).
Setting x = 0, we get 0 + 2y = 24, which gives y = 12. So another point is (0, 12).
Plotting these points and drawing a line through them, we have another boundary line.
Next, we need to shade in the region that satisfies the given conditions. To do this, we can test a point from each region.
For example, let's test the point (0, 0). If we substitute x = 0 and y = 0 into each inequality, we can see if the conditions are met.
For 2x + y ≤ 30:
2(0) + (0) ≤ 30
0 ≤ 30 - This condition is met.
For x + 2y ≤ 24:
0 + 2(0) ≤ 24
0 ≤ 24 - This condition is met.
Since all conditions are met, we shade the entire region below both lines.
Now let's identify the corner points of the feasibility region. These are the points where the two lines intersect.
Solving the equations 2x + y = 30 and x + 2y = 24 simultaneously, we find the point of intersection to be (6, 12).
So, the corner point of the feasibility region is (6, 12).
Finally, we need to find the point(s) (x, y) that maximize the objective function z = 16x + 8y. We can achieve this by evaluating the objective function at each corner point and comparing the results.
Plugging in the coordinates of the corner point into the objective function, we get:
z = 16(6) + 8(12)
= 96 + 96
= 192
Hence, the point (6, 12) maximizes the objective function z = 16x + 8y, with a value of 192.