The given problem is a linear programming problem, and we are tasked with maximizing the objective function $z = 10x + 60y$ under the given constraints:

Objective function:

$z = 10x + 60y$

Constraints:

1. $x + y \leq 200$
2. $x \geq 20$
3. $y \geq 4x$

To solve this, we need to:

1. Graph the constraints to identify the feasible region.
2. Find the vertices of the feasible region.
3. Evaluate the objective function at each vertex to determine the maximum value.

Step 1: Graph the constraints

• The inequality $x + y \leq 200$ represents a line with an intercept at $(200, 0)$ and $(0, 200)$.
• The constraint $x \geq 20$ is a vertical line at $x = 20$.
• The constraint $y \geq 4x$ is a line with a slope of 4, passing through the origin.

Step 2: Find the vertices of the feasible region

We will calculate the intersection points of the boundary lines.

• Intersection of $x + y = 200$ and $y = 4x$:

$x + 4x = 200 \implies 5x = 200 \implies x = 40$ $y = 4(40) = 160$ So, one vertex is $(40, 160)$.

• Intersection of $x = 20$ and $y = 4x$:

$y = 4(20) = 80$ So, another vertex is $(20, 80)$.

• Intersection of $x = 20$ and $x + y = 200$:

$20 + y = 200 \implies y = 180$ So, the third vertex is $(20, 180)$.

Step 3: Evaluate the objective function at each vertex

We now evaluate $z = 10x + 60y$ at each of the vertices:

1. At $(40, 160)$: $z = 10(40) + 60(160) = 400 + 9600 = 10,000$

2. At $(20, 80)$: $z = 10(20) + 60(80) = 200 + 4800 = 5,000$

3. At $(20, 180)$: $z = 10(20) + 60(180) = 200 + 10,800 = 11,000$

Conclusion:

The maximum value of $z$ is $11,000$, which occurs at the point $(20, 180)$.

Let me know if you want more details on the steps or any clarifications!

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Here are 5 related questions for further practice:

1. What happens if the objective function is changed to $z = 20x + 40y$?
2. How would the solution change if the constraint $y \geq 4x$ becomes $y \geq 3x$?
3. What are the implications of removing the constraint $x \geq 20$?
4. Can the feasible region be unbounded for certain values of $x$ and $y$?
5. How do you solve this problem using the Simplex method?

Tip: Always check if the constraints form a closed and bounded region to ensure that a maximum or minimum exists.