Linear Programming Skills Assessment

Formulate a linear inequality for the following scenario: A company produces two products, X and Y. The profit per unit of X is $10 and the profit per unit of Y is $15. The company can produce at most 100 units of X and 150 units of Y. What is the inequality representing the total profit?

• 10x + 15y >= P

• 10x + 15y = P

• 10x + 15y <= P

• 10x - 15y <= P

Identify the feasible region for the following system of inequalities: 2x + 3y ≤ 120, x + y ≤ 60, x ≥ 0, y ≥ 0.

• Feasible region is a parabola opening downwards.

• Feasible region is the triangle bounded by the lines x=0, y=0, and 2x + 3y = 120.

• Feasible region is a rectangle with sides parallel to the axes.

• Feasible region is a circle with center at (30, 30) and radius 30.

Solve the following linear programming problem: Maximize Z = 5x + 3y subject to the constraints 2x + y ≤ 10, x + 3y ≤ 12, x ≥ 0, y ≥ 0.

• The maximum value of Z is 26.

• The maximum value of Z is 15.

• The maximum value of Z is 20.

• The maximum value of Z is 30.

Interpret the optimal solution in the context of the problem: A farmer has 40 acres of land to plant wheat and barley. Each acre of wheat requires 2 hours of labor and $20 of investment, while each acre of barley requires 3 hours of labor and $30 of investment. If the farmer can afford 120 hours of labor and $800 of investment, and the profit per acre of wheat is $100 and barley is $150, what is the optimal mix of crops?

• wheat: 30, barley: 10

• {"wheat": 20, "barley": 20}

• wheat: 15, barley: 25

• wheat: 25, barley: 15

Apply the simplex method to solve the following linear programming problem: Maximize Z = 3x + 2y subject to the constraints x + y ≤ 4, 2x + y ≤ 5, x ≥ 0, y ≥ 0.

• Z = 9, x = 4, y = 1

• Z = 7, x = 3, y = 1

• Z = 11, x = 2, y = 2

• Z = 5, x = 1, y = 2