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:

\[x\ge 0\] \[y\ge 0\] \[\begin{aligned} & 2x+6y\ge 24 \\ & 3x+3y\ge 18 \\ \end{aligned}\]

(a) Write the linear function that describes the total labor cost if a supervisor makes $90 per day and a helper makes $100 per day.

(c) Evaluate the linear function for total labor cost that you found in step (b) at each vertex of the graphed region. The vertices are (0, 6), (3, 3), and (12,0).

(d) Complete the missing portions of this statement:

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