Question

Let the two variables $ x $ and $ y $ satisfy the following conditions: \[ x + y \leq 50, \quad x + 2y \leq 80, \quad 2x + y \geq 20, \quad x, y \geq 0. \] Then the maximum value of $ Z = 4x + 3y $ is:

• A. 120
• B. 170
• C. 200
• D. 210

Hint:

When maximizing or minimizing a linear function subject to constraints, graph the constraints and evaluate the objective function at each corner point of the feasible region.

Answer 1

The Correct Option is C

We are given a system of linear inequalities:
1. \( x + y \leq 50 \), 2. \( x + 2y \leq 80 \), 3. \( 2x + y \geq 20 \), 4. \( x, y \geq 0 \). We are asked to maximize \( Z = 4x + 3y \). 
Step 1: Graph the constraints.
We graph the inequalities to find the feasible region. 
Step 2: Identify the corner points.
By solving the system of equations, we find the corner points of the feasible region. 
Step 3: Calculate \( Z \) at each corner point.
For each corner point, we calculate the value of \( Z = 4x + 3y \). The maximum value occurs at \( x = 40 \) and \( y = 10 \), giving: \[ Z = 4(40) + 3(10) = 160 + 40 = 200. \] Thus, the maximum value of \( Z \) is \( 200 \).