Question

If \( R_1 \) is the feasible region, then find the maximum value of the objective function \( z = 5x + 2y \).

Hint:

To maximize a linear objective function: 
1. Identify all vertices of the feasible region. 
2. Substitute the coordinates of each vertex into the objective function. 
3. Choose the highest value for maximization.

Answer 1

Step 1: The objective function \( z = 5x + 2y \) is maximized at one of the vertices of the feasible region \( R_1 \). The vertices of \( R_1 \) are \( A(0, 50) \), \( B(20, 40) \), and \( C(50, 100) \). 
Step 2: Evaluate \( z \) at each vertex: - At \( A(0, 50) \): \[ z = 5(0) + 2(50) = 100. \] - At \( B(20, 40) \): \[ z = 5(20) + 2(40) = 100 + 80 = 180. \] - At \( C(50, 100) \): \[ z = 5(50) + 2(100) = 250 + 200 = 450. \] 
Step 3: Conclusion: The maximum value of \( z = 5x + 2y \) is 450 at \( C(50, 100) \).