Linear Programming Problem

We are given the objective function , and we need to maximize it subject to the following constraints:
1.
2.
3.
4.
To solve this, we graph the constraints and find the feasible region formed by these inequalities. The vertices of the feasible region are where the lines intersect, as these are the points that will give the maximum or minimum value of .
Step 1: Graph the constraints
The first constraint is , which represents all points to the right of the -axis. The second constraint is , which represents all points above the -axis. The third constraint is , which is the region below the line . The fourth constraint is , which is the region below the line .
Step 2: Find the intersection points
We need to find the points where the constraint lines intersect, as these will be the vertices of the feasible region. 1. Intersection of and : Solve the system of equations: Subtract equation (1) from equation (2): Substitute into equation (1): So, the intersection point is . 2. Intersection of with (y-axis): Substitute into the equation : So, the point is . 3. Intersection of with (x-axis): Substitute into : So, the point is .
Step 3: Calculate at the vertices
We now calculate the value of at each vertex: - At : - At : - At : Thus, the maximum value of is 65 at the point .

To determine the maximum value of the objective function at the given vertices of the feasible region:

1. Evaluate at Origin (0,0):

2. Evaluate at Point A (10,0):

3. Evaluate at Point B (8,4):

4. Evaluate at Point C (0,12):

5. Determine Maximum Value:
Comparing all calculated values:

The maximum value occurs at point C (0,12).

Final Answer:
The maximum value of is .