Question

Maximize \( z = 8000x + 12000y \) subject to constraints: \[ 3x + 4y \leq 60, \quad x + y \leq 30, \quad x \geq 0, \quad y \geq 0. \]

Hint:

In linear programming, the maximum or minimum value of the objective function occurs at one of the corner points of the feasible region.

Answer 1

Step 1: Graph the constraints on the coordinate plane. Plot the inequalities \( 3x + 4y \leq 60 \) and \( x + y \leq 30 \), along with \( x \geq 0 \) and \( y \geq 0 \). 

Step 2: Identify the feasible region formed by the intersection of the inequalities. 

Step 3: The objective function is \( z = 8000x + 12000y \). To maximize \( z \), find the coordinates of the corner points of the feasible region. Step 4: Evaluate \( z \) at each corner point and select the point that gives the highest value of \( z \). Thus, the maximum value of \( z \) is obtained at the appropriate corner point.