Question

The maximum value of \( z = 3x + 4y \), subject to the constraints \( x + y \leq 40, x + 2y \geq 60 \) and \( x, y \geq 0 \) is:

• A. 120
• B. 140
• C. 40
• D. 130

Hint:

In linear programming problems, always check the corner points of the feasible region to find the optimal value of the objective function.

Answer 1

The Correct Option is B

The given problem is an optimization problem subject to the constraints. To find the maximum value of \( z = 3x + 4y \), we first identify the corner points of the feasible region. The corner points are determined by solving the system of constraints: 1. \( x + y \leq 40 \) 2. \( x + 2y \geq 60 \) 3. \( x, y \geq 0 \) By solving the system, we find the following corner points: - \( A = (0, 0) \), \( z = 3(0) + 4(0) = 0 \) - \( B = (40, 0) \), \( z = 3(40) + 4(0) = 120 \) - \( C = (20, 20) \), \( z = 3(20) + 4(20) = 140 \) (Maximum) - \( D = (0, 30) \), \( z = 3(0) + 4(30) = 120 \) Thus, the maximum value of \( z \) is 140 at the corner point \( C = (20, 20) \).