Question

The feasible region of the linear programming problem is determined by the system of inequalities: \[ x + y \leq 6, \quad x \geq 0, \quad y \geq 0. \] What is the maximum value of \( x + y \) in the feasible region?

• A. \( 6 \)
• B. \( 5 \)
• C. \( 4 \)
• D. \( 3 \)

Hint:

When solving linear programming problems, identify the feasible region and evaluate the objective function at the vertices of the region.

Answer 1

The Correct Option is A

The objective is to determine the maximum value of \( x + y \) subject to the constraints: \[ x + y \leq 6, \quad x \geq 0, \quad y \geq 0. \] Step 1: Visualize the Constraints The constraint \( x + y \leq 6 \) defines the region on or below the line \( x + y = 6 \). The constraints \( x \geq 0 \) and \( y \geq 0 \) restrict the feasible region to the first quadrant. The feasible region is a triangle bounded by the line \( x + y = 6 \) and the positive x and y axes. Step 2: Identify the Vertices The vertices of the feasible region are the intersection points: (0, 0), where \( x + y = 6 \) intersects the x-axis at (6, 0), and where \( x + y = 6 \) intersects the y-axis at (0, 6). The vertices are \( (0, 0) \), \( (6, 0) \), and \( (0, 6) \). Step 3: Evaluate the Objective Function at Vertices For each vertex, calculate \( x + y \): - At \( (0, 0) \): \( 0 + 0 = 0 \) - At \( (6, 0) \): \( 6 + 0 = 6 \) - At \( (0, 6) \): \( 0 + 6 = 6 \) Step 4: Determine the Maximum Value The maximum value of \( x + y \) within the feasible region is \( 6 \). This maximum is achieved at the points \( (6, 0) \) and \( (0, 6) \). Answer: The maximum value of \( x + y \) is \( 6 \). This corresponds to option (1).