Problem Statement

You work for a large tech company and have been put in charge of a new division which is tasked with developing cellphones 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 shole cell phone plan and your division with it$.$ 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 $$62,$ and the one deluxe model will see for $$68.$ If we let $x$ represent the number of standard models sold and let $y$ represent the number of deluxe models sold, the revenue function is given by

$R(x,y)=62x+68y$

Again, it is not a sufficient strategy 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 that $460$ 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 make sure that

$3x+4y460$

Constraint $2$: The boss is also a big believer in creating a strong work ethic so the number of phones you can produce must require a minimum of $360$ hours a week so you must also make sure that

$3x+4y360$

Constraint $3$: The boss then informs you that 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 $$20$ and the parts for one deluxe phone cost $$10.$ Since you have a weekly budget of $$1900$ you must also ensure that

$20x+10y1900=>2x+y190$

If we put all of this information together, your task is to maximize

$R(x,y)=62x+68y$

subject to the constraints

$x0,y0$

$3x+4y460,3x+4y360$

and $2x+y190$

$7$

Questions

Graph each of the lines defined by the constraints.

a$.$ Which line from the figure corresponds to $3x+4y=460?$

b$.$ Which line from the figure corresponds to $3x+4y=360?$

c$.$ Which line from the figure corresponds to $2x+y=190?$

Identify the region from the figure below that represents the feasible region defined by the constraints.