Hey Guys, hope you are all ok. So I just need help on 4 and 5. Thanks for the support, all the best!

Question

A construction firm employs two levels of tile installers; craftsmen and apprentice. Craftsmen install 500 square feet of specialty tile, 100 square feet of plain tile, and 100 linear feet of trim in one day. An apprentice installs 100 square feet of specialty tile, and 200 square feet of plain tile, and 100 linear feet of trim in one day. The firm has a one-day job that requires 2000 square feet of specialty tile, 1600 square feet of plain tile, and 1200 linear feet of trim.

Your Task:

Complete each of the following activities. Graph where necessary, and explain them in great detail, so that you can convey the message to the firm telling them what makes the most sense.

Activity ONE:

Let x represent the number of craftsmen, and let y represent the number of apprentices. Write a system of three equations to represent the construction firm’s situation with this job.

Activity TWO:

The construction firm pays craftsmen $ 200 per day and pays apprentices $120 per day.  

1. Write an objective function for the labor costs.
2. Create a system of inequalities to represent the constraints. Graph the feasible region. (DESMOS)
3. Identify the vertices of the feasible region.
4. How many craftsmen and how many apprentices should be assigned to this job so that it can be completed in one day with the minimum labor cost? What is the minimum labor cost?

Activity THREE:  

       Suppose that each apprentice’s wages are increased to $ 150 per day.

1. In this case, how many craftsmen and how many apprentices should be assigned to the job so that it can be completed in one day with the minimum labor cost? What is the minimum labor cost?
2. Do any points in the feasible region call for apprentices but no craftsmen? If so, is this a realistic scenario for the construction firm? What constraint could you add to ensure that every job has at least one craftsmen assigned to it?

Activity FOUR: 

       Suppose that union regulations require at least 1 craftsman for every 3 apprentices on a job.  

1. Which of the vertices of the original feasible region (from activity two) satisfy this new constraint?
2. Write an inequality to represent the new constraint. Modify the feasible region on your graph by adding the boundary line for the new constraint.
3. Using the new constraints and the original labor cost of $ 120 per day for an apprentice, how many craftsmen and how many apprentices should be assigned to the job so that it can be completed in one day with the minimum labor cost? What is the minimum labor cost?

Activity FIVE:

       Suppose you are asked to re-do the tiles in your classroom. Write up a detailed proposal, including everything that would be necessary to get the job done in one day.

Guiding questions:

1. What will you need to get the job done?
2. Who will you need to hire?

Answer 1

Activity ONE:

Let x represent the number of craftsmen, and let y represent the number of apprentices. We can write the following system of equations to represent the construction firm's situation with this job:

Craftsmen:

500x + 100y = 2000 (specialty tile)

100x + 200y = 1600 (plain tile)

100x + 100y = 1200 (trim)

Activity TWO:

Objective function for labor costs:

Cost = 200x + 120y

Constraints:

x, y ≥ 0 (non-negativity)

Graphing the feasible region on DESMOS will help visualize the solution space based on the constraints.

Identify the vertices of the feasible region and find the minimum labor cost by evaluating the objective function at each vertex.

Activity THREE:

Suppose each apprentice's wages are increased to $150 per day.

Repeat the steps from Activity TWO with the new wage rate for apprentices.

Check if any points in the feasible region require apprentices but no craftsmen. Evaluate if this is a realistic scenario for the construction firm. To ensure that every job has at least one craftsman assigned to it, you can add the constraint x ≥ 1.

Activity FOUR:

Suppose union regulations require at least 1 craftsman for every 3 apprentices on a job.

Identify which vertices of the original feasible region satisfy this new constraint.

Write an inequality to represent the new constraint: x ≥ (1/3)y

Modify the feasible region on the graph by adding the boundary line for the new constraint.

Using the new constraints and the original labor cost of $120 per day for an apprentice, find the minimum labor cost by evaluating the objective function at each feasible vertex.

Activity FIVE:

When re-doing the tiles in your classroom, the detailed proposal should include:

- Calculating the total square footage of specialty tile, plain tile, and linear feet of trim required for the job.

- Determining the number of craftsmen and apprentices needed based on their productivity rates per day (as given in the problem statement).

- Estimating the labor cost based on the wage rates provided by the construction firm.

- Identifying the necessary tools, equipment, and materials for the tile installation.

- Outlining the project timeline and sequencing of tasks.

- Considering safety measures and precautions during the installation process.

- Determining if any additional skilled labor, such as a supervisor or project manager, is required to oversee the job.

For hiring, you would need to hire craftsmen and apprentices based on the determined quantities needed for the job.

Hope this helps.