Let's break down and solve the first problem systematically using linear programming techniques, as the prompt suggests.

Given Problem:

• A carpenter makes large and small bookcases.
• Time to make a large bookcase: 4 hours
• Time to make a small bookcase: 2 hours
• Profit per large bookcase: $35
• Profit per small bookcase: $20
• The carpenter can spend 32 hours per week on bookcases.
• Must make at least 2 large and 4 small bookcases per week.

Steps to solve:

a) Identify the function to maximize or minimize

The goal is to maximize profit.

b) Define your variables

Let:

• $x$ = number of large bookcases
• $y$ = number of small bookcases

c) Write the objective function

The objective is to maximize the profit: $P = 35x + 20y$ where $35x$ is the profit from the large bookcases and $20y$ is the profit from the small ones.

d) Write the system of inequalities representing the constraints

1. Time constraint: The total time spent making bookcases should not exceed 32 hours: $4x + 2y \leq 32$
2. Minimum bookcase constraints: The carpenter must make at least 2 large and 4 small bookcases: $x \geq 2$ $y \geq 4$
3. Non-negative constraints: The number of bookcases can’t be negative: $x \geq 0, \quad y \geq 0$

e) Graph the feasibility region

To graph the feasible region, we need to plot the following inequalities:

1. $4x + 2y \leq 32$
2. $x \geq 2$
3. $y \geq 4$

We will identify the points where these constraints intersect and form the boundaries of the feasible region.

f) Determine the vertices of the feasible region

By solving the system of equations for the points where the constraints intersect, we can find the vertices of the feasible region.

Let me now solve the system and find these vertices, and then calculate the maximum profit based on these points.The carpenter should make 2 large bookcases and 12 small bookcases to maximize his profit. The maximum profit is $310.

Would you like to go into more detail or ask any questions? Here are five related questions to extend this topic:

1. What happens to the solution if the carpenter's total time is increased?
2. How would the solution change if the profit on the small bookcases increased?
3. What is the importance of the vertices of the feasibility region in linear programming?
4. Can you explain how the simplex method works for maximizing profit in linear programming?
5. How would you graphically represent this problem?

Tip: Linear programming problems like this one often involve identifying constraints and finding an optimal solution based on feasible points where constraints intersect.