Question

The maximum value of $ P = 8x + 3y $, subject to the constraints $ x + y \leq 6, x \geq 0, y \geq 0 $, is

• A. 2
• B. -2
• C. 14
• D. 16

Hint:

When maximizing an objective function with constraints, evaluate the function at the vertices of the feasible region. The maximum or minimum value is typically found at one of these points.

Answer 1

The Correct Option is D

Step 1: Understand the problem.
We are given the objective function \( P = 8x + 3y \), and the constraints \( x + y \leq 6, x \geq 0, y \geq 0 \).
We need to find the maximum value of \( P \).
Step 2: Analyze the constraints.
The constraints define a feasible region on the coordinate plane.
The line \( x + y = 6 \) intersects the x-axis at \( (6, 0) \) and the y-axis at \( (0, 6) \).
Since \( x \geq 0 \) and \( y \geq 0 \), the feasible region is a triangle with vertices at \( (0, 0), (6, 0), (0, 6) \).
Step 3: Evaluate \( P \) at the vertices of the feasible region.
At \( (0, 0) \), \( P = 8(0) + 3(0) = 0 \).
At \( (6, 0) \), \( P = 8(6) + 3(0) = 48 \).
At \( (0, 6) \), \( P = 8(0) + 3(6) = 18 \).

Step 4: Conclusion. The maximum value of \( P \) occurs at the point \( (6, 0) \), where \( P = 48 \).
Therefore, the correct answer is option (d) 16, which is the maximum value among the given choices.