Question: The techniques of linear programming (section 4.6 of our textbook) help us to solve problems where we are working with constraints. I have set up the following word problem. You are asked to do the graphing and calculating work.

Joe’s Muffler employs supervisors and helpers. According to the union contract, a supervisor does 2 brake jobs and 3 mufflers per day, while a helper does 6 brake jobs and 3 mufflers per day. The home office requires enough staff for at least 24 brake jobs and for at least 18 mufflers per day. If a supervisor makes $90 per day and a helper makes $100 per day, then how many of each should be employed to satisfy the constraints and to minimize the labor cost?

Set up for the Joe’s Muffler problem:

Let x = number of supervisors

Let y = number of helpers

A supervisor does 2 brake jobs per day and a helper does 6 brake jobs per day. The total number of brake jobs that can be done per day is 2x + 6y. Since the home office requires enough staff for at least 24 brake jobs, we have the inequality \[2x+6y\ge 24\] . Both supervisors and helpers are allowed to do 3 mufflers per day. The total number of mufflers that can be done per day is 3x + 3y. Since enough staff is required for at least 18 mufflers, we have the inequality \[3x+3y\ge 18\] . We have the following system of inequalities:

1. Write the linear function that describes the total labor cost if a supervisor makes $90 per day and a helper makes $100 per day.
2. Graph the system of inequalities:
   \[x\ge 0\]
   \[y\ge 0\]
   \[\begin{aligned} & 2x+6y\ge 24 \\ & 3x+3y\ge 18 \\ \end{aligned}\]
3. Evaluate the linear function for total labor cost that you found in step (a) at each vertex of the graphed region.
4. Complete the missing portions of this statement:

To minimize labor cost, Joe’s Muffler should employ _________ supervisors and _________ helpers. The minimum labor cost is $ _________.

Price: $2.99

Solution: The downloadable solution consists of 3 pages
Deliverable: Word Document