The problem requires us to analyze a system of inequalities and determine the feasible region's corner points, then evaluate the objective function $M = 3x + 8y$ at those points to find the maximum and minimum values.

Here are the steps to approach this:

Step 1: Interpret the Inequalities and Identify Constraints

The given inequalities in the system are:

1. $20x + 40y \geq 200$
2. $15x + 15y \geq 135$
3. $16x \geq 48$
4. $y \geq 0$

Let's rewrite these inequalities in terms of lines (by turning them into equalities) and solve for the intercepts, which will help identify the corner points.

Step 2: Convert Inequalities to Equations and Graph

The boundary lines of each inequality are:

1. $20x + 40y = 200$, which simplifies to $x + 2y = 10$.

   • This line has an x-intercept at $(10, 0)$ and a y-intercept at $(0, 5)$.

2. $15x + 15y = 135$, which simplifies to $x + y = 9$.

   • This line has an x-intercept at $(9, 0)$ and a y-intercept at $(0, 9)$.

3. $16x = 48$, which simplifies to $x = 3$.

   • This is a vertical line at $x = 3$.

4. $y = 0$.

   • This is the x-axis, where $y = 0$ for all values of $x$.

Step 3: Identify the Feasible Region

The feasible region is where all inequalities overlap on the graph. It appears to be a bounded polygon, so we can determine the corner points by finding the intersections of these lines.

Step 4: Calculate the Corner Points

Using the equations for the boundary lines, we can find the intersection points:

1. Intersection of $x + 2y = 10$ and $x + y = 9$:

   • Solving these simultaneously, we get $x = 8$ and $y = 1$.
   • Point: $(8, 1)$

2. Intersection of $x + 2y = 10$ and $x = 3$:

   • Substituting $x = 3$ into $x + 2y = 10$, we get $y = 3.5$.
   • Point: $(3, 3.5)$

3. Intersection of $x + y = 9$ and $x = 3$:

   • Substituting $x = 3$ into $x + y = 9$, we get $y = 6$.
   • Point: $(3, 6)$

4. Intersection of $x = 3$ and $y = 0$:

   • Point: $(3, 0)$

Step 5: Evaluate the Objective Function at Each Corner Point

The objective function is $M = 3x + 8y$. Let’s calculate $M$ for each corner point:

1. At $(8, 1)$: $M = 3(8) + 8(1) = 24 + 8 = 32$

2. At $(3, 3.5)$: $M = 3(3) + 8(3.5) = 9 + 28 = 37$

3. At $(3, 6)$: $M = 3(3) + 8(6) = 9 + 48 = 57$

4. At $(3, 0)$: $M = 3(3) + 8(0) = 9$

Step 6: Determine the Maximum and Minimum Values

• Maximum value of $M$: 57 at $(3, 6)$.
• Minimum value of $M$: 9 at $(3, 0)$.

Summary

• Corner Points: $(8, 1), (3, 3.5), (3, 6), (3, 0)$
• Maximum value of $M$: 57 at $(3, 6)$
• Minimum value of $M$: 9 at $(3, 0)$

Would you like a more detailed explanation for any specific part, or have any further questions?

Here are some related questions to explore further:

1. How would changing the objective function impact the maximum and minimum values?
2. What if an additional constraint was added? How would it change the feasible region?
3. How can we interpret these results in a real-world context?
4. What is the significance of the feasible region in a linear programming problem?
5. How would the solution change if we modified one of the inequality signs?

Tip: In linear programming, always verify the feasible region's corner points since they determine the possible optimal values for the objective function.