Maximize z=16x + 8y subject to:
2x + y<30
x +2y<24
x>0
y>0
Why did the mathematician become a farmer?
Because he wanted to maximize his crops and minimize his equations!
To maximize the objective function z=16x+8y subject to the given constraints, we can use the graphical method or the simplex method. Let's solve it using the graphical method.
1. Plot the inequalities on a graph:
First, plot the two equality lines by converting them to equations:
2x + y = 30 (equation 1)
x + 2y = 24 (equation 2)
Now, plot the lines on the graph.
2. Determine the feasible region:
The feasible region is the area on the graph that satisfies all the constraints. In this case, the feasible region is the area below the line 2x + y = 30, below the line x + 2y = 24, and in the first quadrant (where x > 0 and y > 0).
Shade the feasible region on the graph.
3. Find the corner points of the feasible region:
The corner points of the feasible region are the points where the boundary lines intersect.
Label the corner points as A, B, C, and D.
4. Evaluate z at each corner point:
Substitute the x and y values of each corner point into the objective function z=16x+8y to find the value of z at each corner point.
For corner point A: zA = 16(0) + 8(0) = 0
For corner point B: zB = 16(0) + 8(12) = 96
For corner point C: zC = 16(8) + 8(2) = 144
For corner point D: zD = 16(6) + 8(0) = 96
5. Find the maximum value of z and the corresponding values of x and y:
Compare the values of z at each corner point and identify which one is the maximum.
The maximum value of z is 144, which occurs at corner point C (x=8, y=2).
Therefore, the maximum value of z=144 occurs when x=8 and y=2, subject to the given constraints.
To maximize the objective function z = 16x + 8y subject to the given constraints, we can use the method of linear programming. Here's a step-by-step explanation of how to solve it:
1. Graph the constraints: The first step is to graph the given constraints on a coordinate plane to see the feasible region. Let's start with the inequality 2x + y < 30:
- Rewrite it as an equation: 2x + y = 30
- Plot the line: To plot this line, you need two points. Let's choose two easy points, (0, 30) and (15, 0).
- Shade the feasible region: Since the inequality is '<', shade the area below the line.
Repeat the same steps for the second constraint x + 2y < 24.
2. Find the feasible region: The feasible region is the overlapping shaded area from the graphs of the two constraints. This region satisfies all the given conditions.
3. Identify the corner points: Corner points are the vertices or intersections of the lines forming the feasible region. Locate all the corner points in the feasible region.
4. Evaluate the objective function at each corner point: Plug the coordinates of each corner point into the objective function z = 16x + 8y to find the corresponding values of z.
5. Determine the maximum value of z: Compare the values of z obtained from each corner point and find the maximum value. The corner point that yields the maximum value of z is the optimal solution.
Note: In this case, since the feasible region is bounded and the objective function is linear, the maximum value of z will occur at one of the corner points.
I hope this explanation helps you understand how to solve the given linear programming problem.