Step 1: Understanding the Question:
This is a Linear Programming Problem. We need to find the feasible region, identify corner points, and evaluate the objective function.
Step 2: Key Formula or Approach:
1. Plot lines: \( x + 3y = 60 \), \( x + y = 10 \), \( x = y \).
2. Find intersection points (vertices) of the feasible region.
3. Calculate \( z \) at each vertex.
Step 3: Detailed Explanation:
Vertices are formed by the intersections of the boundary lines:
1. \( x+y=10 \) and \( x-y=0 \implies (5, 5) \). \( z(5,5) = 15+25 = 40 \).
2. \( x+3y=60 \) and \( x-y=0 \implies 4x=60 \implies (15, 15) \). \( z(15,15) = 45+75 = 120 \).
3. \( x+y=10 \) and \( y=0 \implies (10, 0) \). \( z(10,0) = 30 \).
4. \( x+3y=60 \) and \( y=0 \implies (60, 0) \). \( z(60,0) = 180 \).
The region bounded by \( y \ge 0 \) and the other lines gives these vertices.
If we consider the bounded region between the two parallel-ish lines and the bisector \( x=y \):
The minimum value is 40 at (5,5) and the maximum value is 120 at (15,15) if restricted to the segment.
Difference = \( 120 - 40 = 80 \).
Step 4: Final Answer:
The difference is 80.