Painting: Need help with getting equations!

Michael Thomas, the manager of a paint store, is mixing paint for a spring sale. There are 32 units of yellow dye, 54 units of brown dye, and an unlimited supply of base paint available. Mr.Thomas plans to mix as many gallons as possible of Autumn Wheat and Harvest Brown paint. Each gallon of Autumn Wheat requires 4 units of yellow dye, and 1 unit of brown dye. Each gallon of Harvest Brown paint requires 1 unit of yellow dye and 6 units of brown dye. Find the maximum number of gallons of paint that Mr.Thomas can mix.

Question

Painting: Need help with getting equations!

Michael Thomas, the manager of a paint store, is mixing paint for a spring sale. There are 32 units of yellow dye, 54 units of brown dye, and an unlimited supply of base paint available. Mr.Thomas plans to mix as many gallons as possible of Autumn Wheat and Harvest Brown paint. Each gallon of Autumn Wheat requires 4 units of yellow dye, and 1 unit of brown dye. Each gallon of Harvest Brown paint requires 1 unit of yellow dye and 6 units of brown dye. Find the maximum number of gallons of paint that Mr.Thomas can mix.

openstudyuser · Answer

Yellow Dye= x Brown dye=y

anonymous · Answer

@dumbcow

anonymous · Answer

So first of all $x\leq 32 $ and $y \leq 54$

openstudyuser · Answer

So when he says there are unlimited, we put greater than or equal to?

anonymous · Answer

Well the dye is mixed with the base paint.  The point is we won't run out of base paint, but will run out of dye.

anonymous · Answer

We can ignore the base paint, basically.

openstudyuser · Answer

So how did you get the greater than or equal to sign for the both of them?

anonymous · Answer

$A = 4x+y$ and $H=1x+6y$

anonymous · Answer

Well, it's okay NOT to run out of dye.

anonymous · Answer

So if we use less than how much we have, we are still ok.

openstudyuser · Answer

For this one, we have two major equations?

anonymous · Answer

yeah but we may combine them too.

openstudyuser · Answer

Isnt Autumn Wheat the yellow paint and Harvest brown the brown one right?

anonymous · Answer

Yeah.

openstudyuser · Answer

"Brown paint. Each gallon of Autumn Wheat requires 4 units of yellow dye, and 1 unit of brown dye. Each gallon of Harvest Brown paint requires 1 unit of yellow dye and 6 units of brown dye"
Thats the stuff i was mainly confused on.

anonymous · Answer

That is represented with my A, and H equations.

openstudyuser · Answer

Basically, what they are saying is that to make a Autumn Wheat paint, they need to mix 4 of the yellow and 1 of the brown. And to make Harvest Brown, they need 1 yellow and 6 brown?

anonymous · Answer

Yes.

openstudyuser · Answer

You would have to graph those A and H equations?

anonymous · Answer

I don't think so.

anonymous · Answer

We have many unknowns.

openstudyuser · Answer

Yea, cuz we need one more equation to graph, if we only graph, $$x \le32$$   and $$y \le54$$
Wouldnt that make the graph unbounded?

anonymous · Answer

We don't know how many of A or H we'll make, so we need a quantity for that.
The total gallons, $g$ is gonna be:
$ g= a(4x+y) + h(x+6y)$

anonymous · Answer

Well we also know we can't get into negatives, so it's more  like $$0\leq x \leq 32$$$$0\leq y \leq 54 $$

anonymous · Answer

But that's a lot of potential points to consider, soooo...

openstudyuser · Answer

Iam lost. lol.

anonymous · Answer

I think it's one of those problems where you just have to try a bunch of points out.

openstudyuser · Answer

In which equations?

anonymous · Answer

Heh, that's the thing, even when you know what point you're going to use, you still have to decide whether you want to make Autumn or Harvest... let me think on this a bit.

openstudyuser · Answer

Okay, im just very confused on this concept lol.

anonymous · Answer

I mean, if we only make Autumn paint, how many gallons can we make, if we only make Harvest paint, how many can we make?  If we try for half and half, how many do we make?

openstudyuser · Answer

A whole? lol i dunno?

anonymous · Answer

Hmmm, actually I think I might has confused myself too, and messed up one of the equations.

anonymous · Answer

Okay, so if we only made autumn we'd make 8, with a ton of brown dye remaining.  If we only make harvest we have 9 with a ton of yellow dye remaining.

anonymous · Answer

Catch my drift?

openstudyuser · Answer

Ummmmm...no not really lol.

anonymous · Answer

AW pain takes 4 yellow, we only have 32 yellow so 32/4 = 8 is the most we could make of it.

anonymous · Answer

Doing so would only take up 8 * 1 = 8 brown, and we'd have 54 - 8 = 46 brown left.

anonymous · Answer

For HB, it takes 6 brown.  54 / 6 = 9 we could make total, with 32 - 9 = 23 yellow pain remaining.

openstudyuser · Answer

If that were the case wouldnt$$x \ge32$$ and $$y \ge54$$?

anonymous · Answer

No, they are less then or equal.

anonymous · Answer

saying they are greater than or equal would mean we could use as much as we wanted.

openstudyuser · Answer

But wouldnt we have limited paints?

anonymous · Answer

we have limited dyes, but unlimited base paint.  I'm talking about dye.

openstudyuser · Answer

So shouldnt it be less than equal to?

anonymous · Answer

it is

anonymous · Answer

Hold on, I figured out what we've been doing wrong....
Our variables should have been how much AW and HB we make.

openstudyuser · Answer

So what would it have to equal to?

dumbcow · Answer

i know this is a little late but here is a nice concise solution:
Let A be gallons of Autumn yellow , H be gallons of Harvest brown

-you need 2 restraint equations: 1 for yellow dye and 1 for brown dye
yellow:
$$4A + H \le 32$$
where "4A+H" represents total amount of yellow dye used 
brown:
$$A + 6H \le 54$$
-Graph these 2 equations to see possible region
|dw:1349578303057:dw|
total gallons = A+H
solve for point where lines intersect to obtain maximum
A=6
H=8

--> max gallons = 14