Question

Maximize P = 5x + 3y subject to x + y < 80, 3x < 90, x > 0, and y > 0. Fill in the values of each variable (including the slack variables u and v).

A) x = 10, y = 70
B) x = 30, y = 50
C) x = 20, y = 60
D) x = 40, y = 40

Answer 1

After analyzing the options and computing P for each, option D) x = 40, y = 40 was found to be the maximizing solution with P = 320, which is the highest of all the given options.

Step-by-step explanation:

To maximize the objective function P = 5x + 3y, we need to consider the given constraints.

The constraints are:

• x + y < 80
• 3x < 90
• x > 0
• y > 0

We can add slack variables u and v to turn the inequalities into equations:

• x + y + u = 80
• 3x + v = 90

To identify the optimal solution, we can graph these constraints and look for the feasible region. We will then find the corner points of this region and evaluate the objective function P at these points.

Now, let's analyze the provided options:

A) x = 10, y = 70: P = 5(10) + 3(70) = 50 + 210 = 260

B) x = 30, y = 50: P = 5(30) + 3(50) = 150 + 150 = 300

C) x = 20, y = 60: P = 5(20) + 3(60) = 100 + 180 = 280

D) x = 40, y = 40: P = 5(40) + 3(40) = 200 + 120 = 320, which is the maximum value of P among the options given.