Question

The maximum value of \( Z = 2x + y \) subject to the constraints \[ x + y \leq 35, \quad x \geq 0, \quad y \geq 0 \] is.

• A. 35
• B. 105
• C. 70
• D. 140

Hint:

For linear programming problems, the maximum or minimum values often occur at the vertices of the feasible region.

Answer 1

The Correct Option is C

We are given the constraints \( x + y \leq 35 \), \( x \geq 0 \), and \( y \geq 0 \). The objective function to maximize is \( Z = 2x + y \). To maximize \( Z \), we consider the boundary condition where \( x + y = 35 \). Substituting \( y = 35 - x \) into \( Z = 2x + y \), we get: \[ Z = 2x + (35 - x) = x + 35. \] To maximize \( Z \), we set \( x \) to its maximum value, which is 35 (as \( x + y = 35 \)). Substituting \( x = 35 \) into \( Z \): \[ Z = 35 + 35 = 70. \] Thus, the maximum value of \( Z \) is 70.