Question

Minimize z = 60a + 50b, subject to the constraints: 10a + 20b ≤ 200, 8a + 5b ≤ 80, a ≥ 2, b ≥ 5. Solve this linear program graphically and determine the optimal quantities of a, b, and the value of z. 2x1 – x2 < 200, x1 < 150, x1, x2 > 0.

Answer 1

To solve this linear program graphically, graph the inequalities, plot the constraints, find the corner points of the feasible region, substitute the coordinates into the objective function, and compare the values to determine the optimal quantities and value.

Step-by-step explanation:

To solve this linear program graphically, we will find the feasible region and then determine the coordinates of the corner points of the region. We will substitute these coordinates into the objective function to find the optimal quantities of a and b, as well as the value of z.

1. Graph the inequalities 10a + 20b ≤ 200 and 8a + 5b ≤ 80, and shade the region that satisfies both inequalities.
2. Plot the constraints a ≥ 2 and b ≥ 5 on the graph.
3. Identify the corner points of the feasible region by finding the intersections of the lines.
4. Substitute the coordinates of each corner point into the objective function z = 60a + 50b to find the value of z.
5. Compare the values of z at each corner point to determine the optimal value of z and the corresponding values of a and b.