### Problem 3

#### Problem Statement

You work for a large tech company and have been put in charge of a new division tasked with developing cell phones that could potentially compete with the iPhone. There are two models of phones your division will produce: the standard and the deluxe model. Your supervisor tells you that if this new division does not prove profitable during the first quarter, the owner of the tech company may scrap the whole cell phone plan, along with your division. Therefore, it is up to you to generate the most possible revenue from the division.

The company informs you that one standard model will sell for [tex]$\$[/tex]62[tex]$, and one deluxe model will sell for $[/tex]\[tex]$68$[/tex]. If we let [tex]x[/tex] represent the number of standard models sold and [tex]y[/tex] represent the number of deluxe models sold, the revenue function is given by

[tex]
R(x, y) = 62x + 68y
[/tex]

It is not sufficient to simply sell as many phones as possible because there are some constraints in the manufacturing process that you must be aware of.

Constraint 1: The owner has given you 8 workers and stated they must work no more than 460 total hours in a week. Since the standard model takes 3 hours to make, and the deluxe model takes 4 hours to make, you need to ensure that

[tex]
3x + 4y \leq 460
[/tex]

Constraint 2: The boss is also a big believer in creating a strong work ethic, so the number of phones you produce must require a minimum of 360 hours a week. Therefore, you must also ensure that

[tex]
3x + 4y \geq 360
[/tex]

Constraint 3: The company has certain deals with electronic companies in place and can acquire the parts required to produce the deluxe phones cheaper than the materials required for the standard phone. The parts for one standard phone cost [tex]$\$[/tex]20[tex]$ and the parts for one deluxe phone cost $[/tex]\[tex]$10$[/tex]. Since you have a weekly budget of \$1900, you must ensure that

[tex]
20x + 10y \leq 1900 \Rightarrow 2x + y \leq 190
[/tex]

Putting all this information together, your task is to maximize

[tex]
R(x, y) = 62x + 68y
[/tex]

subject to the constraints:

[tex]
\begin{array}{c}
x \geq 0, \quad y \geq 0 \\
3x + 4y \leq 460, \quad 3x + 4y \geq 360 \\
2x + y \leq 190
\end{array}
[/tex]

#### Questions

1. Graph each of the lines defined by the constraints.
a. Which line from the figure corresponds to [tex]3x + 4y = 460[/tex]?
b. Which line from the figure corresponds to [tex]3x + 4y = 360[/tex]?
c. Which line from the figure corresponds to [tex]2x + y = 190[/tex]?

2. Identify the region from the figure below that represents the feasible region.