Help me out with this linear programming problem please! I will post the question as a screenshot below!

- dinorap1

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- dinorap1

This is the question I need help with. Please help!

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- jim_thompson5910

`The production of y can exceed x by no more than 200` translates to what inequality?

- dinorap1

That's what's stumping me here... I'm not sure how to write that one out.

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## More answers

- jim_thompson5910

it would translate to `y <= x+200`
basically `x+200` is the ceiling or max value for y. It can't go any higher

- jim_thompson5910

also, it's implicitly stated that `x >= 0` and `y >= 0` since x and y are the number of products made

- dinorap1

Oh okay. So now all I have to do is write it out on a graph and plug the vertices into the equation?

- jim_thompson5910

so what you need to do is graph this system of inequalities
\[\Large \begin{cases}y \le x+200\\ x+2y \le 1600\\ x \ge 0\\ y \ge 0\end{cases}\]

- jim_thompson5910

find the shaded feasible region and the corner points of that region. Then plug each corner point into the profit function

- dinorap1

Oh wow, thanks for writing it like that! That helps a lot! :D
I have a graphing tool that does it for me. Maybe you can look at it after I'm done to see if I'm doing it right...

- jim_thompson5910

sure I can do that

- dinorap1

Alright I'm almost done graphing it...

- jim_thompson5910

ok

- dinorap1

Here's what it looks like...

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- jim_thompson5910

don't forget x >= 0 and y >= 0
when you graph all 4 inequalities and focus on the region where all shaded regions overlap, you'll get this

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- dinorap1

So my graph doesn't look right yet...?

- dinorap1

Now it looks like this. :/

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- jim_thompson5910

it's a mess when you do all 4 graphs at the same time
https://www.desmos.com/calculator/wpwycrhycx
but with the link above, you can turn on/off graphs so you can see them individually or grouped

- jim_thompson5910

yeah that looks better. if you focus on the region where all 4 graphs have in common, you'll have this region here

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- dinorap1

That looks much better than my graph. I still know the general area where all the vertices are, so I can probably deal with it looking messy like this... is my graph still accurate besides that?

- jim_thompson5910

each corner point is found by finding the intersection of the lines

- jim_thompson5910

example: (0,200) is found by finding the intersection of the lines `x = 0` and `y=x+200`

- dinorap1

I'm getting a little confused... x>=0 isn't a line, is it? It just prevents x from equalling 0 I thought...

- dinorap1

Same with y>=0

- jim_thompson5910

`x = 0` is the boundary line to `x >= 0`

- jim_thompson5910

`x = 0` is the y axis
`x >= 0` means you shade to the right of the y axis

- dinorap1

Okay I think the answer is C, which is: (1600,0)
Is that right?

- dinorap1

I need to go really soon

- jim_thompson5910

yep the profit is P = 21500 at (x,y) = (1600,0) which is the max profit

- dinorap1

Okay I'm about to submit it then... hope I don't get a crappy grade! I've been working on this for hours!

- dinorap1

I got a 3/3! Thank you for helping! You really helped me understand this better! :)

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