Step-by-Step Solution

1. Graph the Constraints:

We need to graph the following inequalities:

• $x + y \le 200$
• $x \ge 20$
• $y \ge 4x$
• $y \ge 0$

2. Identify the Feasible Region:

The feasible region is the area that satisfies all the inequalities simultaneously. This region is bounded by the lines $x+y=200$, $x=20$, and $y=4x$.

3. Find the Corner Points:

The corner points of the feasible region are the points where the boundary lines intersect. We need to find these points:

• Intersection of $x=20$ and $y=4x$: $x=20$, $y=4(20) = 80$. So, the point is $(20, 80)$.
• Intersection of $x=20$ and $x+y=200$: $x=20$, $20+y=200$, $y=180$. So, the point is $(20, 180)$.
• Intersection of $y=4x$ and $x+y=200$: $x+4x=200$, $5x=200$, $x=40$, $y=4(40)=160$. So, the point is $(40, 160)$.

4. Evaluate the Objective Function at the Corner Points:

We need to evaluate $z = 500x + 400y$ at each corner point:

• At $(20, 80)$: $z = 500(20) + 400(80) = 10000 + 32000 = 42000$
• At $(20, 180)$: $z = 500(20) + 400(180) = 10000 + 72000 = 82000$
• At $(40, 160)$: $z = 500(40) + 400(160) = 20000 + 64000 = 84000$

5. Determine the Minimum Value:

The minimum value of the objective function is the smallest value obtained in the previous step. In this case, the minimum value is $42000$ at the point $(20, 80)$.