Answer:
A. z = 225
B. The maximum value of z occurs at the points (22, 23) and (45, 0).
Explanation:
To solve the linear programming problem using graphical methods, we need to graph the inequalities and find the feasible region. Then, we can identify the corner points of the feasible region and evaluate the objective function at each of these points to determine the maximum value of z.
Objective function: z = 5x + 5y
Constraints:
- 10x + 6y ≥ 150
- 13x - 13y ≥ -13
- x + y ≤ 45
- x ≥ 0
- y ≥ 0
Graph each of these inequalities by first plotting the corresponding boundary line, and then shading in the appropriate region.
Rearrange the first three inequalities to isolate y:

Graph the inequalities.
- If the inequality sign is ≥, draw a solid line and shade above the line.
- If the inequality sign is ≤, draw a solid line and shade below the line.
The feasible region is the region that is shaded by all of the inequalities.
Please see the attached graph.
A bounded feasible region may be enclosed in a circle and will have both a maximum value and a minimum value for the objective function. Therefore, as the feasible region for the given constraints is bounded, there is a maximum value of z.
The feasible region is bounded by the corner points:
- (9, 10)
- (22, 23)
- (45, 0)
- (15, 0)
Evaluate the objective function z = 5x + 5y at each of these corner points:
Point (9, 10): z = 5(9) + 5(10) = 95
Point (22, 23): z = 5(22) + 5(23) =225
Point (45, 0): z = 5(45) + 5(0) = 225
Point (15, 0): z = 5(15) + 5(0) = 75
Therefore, the maximum value of z is 225, which occurs at the corner points (22, 23) and (45, 0).