Question

solve the following linear program using the graphical solution procedure: max 5a 5b s. t. 1a ≤ 100 1b ≤ 80 2a 4b ≤ 400 a, b ≥ 0

Answer 1

To graphically solve the provided linear program, plot the constraints on a graph, identify the feasible region, and find the point which maximizes the objective function within this region.

Step-by-step explanation:

To solve the linear program graphically, we plot the constraints on a graph and find the feasible region. The constraints are 1) a ≤ 100, 2) b ≤ 80, and 3) 2a + 4b ≤ 400. Both a and b must also be greater than or equal to 0, which confines our feasible region to the first quadrant of the coordinate plane.

The objective function to maximize is 5a + 5b. We need to plot the line corresponding to the objective function and move it parallel until it reaches the farthest point still within the feasible region (while still being parallel to its original position), that's the point that maximizes the objective function without violating any constraints.

Steps for Graphical Solution

1. Plot the inequality a ≤ 100. This is a vertical line intersecting the x-axis (a-axis) at a = 100.
2. Plot the inequality b ≤ 80. This is a horizontal line intersecting the y-axis (b-axis) at b = 80.
3. Plot the inequality 2a + 4b ≤ 400. To do this, find the intercepts by setting a to 0 to find the b intercept, and b to 0 to find the a intercept.
4. Draw the feasible region, which is the area that satisfies all the inequalities at once.
5. Find the point in this region that maximizes the objective function 5a + 5b.

After performing these steps, the corner points of the feasible region should be evaluated in the objective function to find the maximum value.