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A carpenter makes bookcases in 2 sizes, large and small. It takes 4 hours to make a large bookcase

and 2 hours to make a small one. The profit on a large bookcase is $35 and on a small bookcase is $20.

The carpenter can spend only 32 hours per week making bookcases and must make at least 2 of the large

and at least 4 of the small each week. How many small and large bookcases should the carpenter make to

maximize his profit? What is his profit?

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.

A carpenter makes bookcases in 2 sizes, large and small. It takes 4 hours to make a large bookcase and 2 hours to make a small one. The profit on a large bookcase is $35 and on a small bookcase is $20. The carpenter can spend only 32 hours per week making bookcases and must make at least 2 of the large and at least 4 of the small each week. How many small and large bookcases should the carpenter make to maximize his profit? What is his profit?

Solve a linear programming problem where a carpenter makes large and small bookcases. Learn how to maximize profit with constraints of time and minimum production. Suitable for high school students studying linear programming.

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