Final answer:
To find the maximum value of z = 3x + 4y with the given constraints, graph the inequalities on a coordinate plane and find the feasible region. The maximum value of z = 3x + 4y is 80, which occurs at the corner point (x, y) = (12, 8).
Step-by-step explanation:
To find the maximum value of z = 3x + 4y with the given constraints, we need to graph the inequalities on a coordinate plane and find the feasible region. The feasible region is the region that satisfies all the constraints.
We start by graphing the lines x + y = 10, x + 3y = 72, and 10x + 3y = 180. Then, we shade the region that satisfies the corresponding inequality. Finally, we find the corner point within the feasible region that maximizes z = 3x + 4y.
The maximum value of z = 3x + 4y is 80, which occurs at the corner point (x, y) = (12, 8).