an open box with no lid has a rectangular base. The height is equal to the shortest side of the base. What are the dimensions of the box if the volume is 208 cubic cm and the surface area is 188 square cm. **i only have two equations below that make sense. how do i start?**

- anonymous

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- anonymous

The equations i know: vol=LWH=xxy=208 and
Area = X^2+xy=188

- zepdrix

Hmm let's draw it a sec.

- zepdrix

|dw:1377983622808:dw|

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## More answers

- zepdrix

I think your surface area equation is a little off for this one.
Let's check.

- zepdrix

|dw:1377983709076:dw|So we have one of these panels which has area x^2.

- zepdrix

|dw:1377983760142:dw|And then we have 4 of these panels right?
Which each have area xy.

- zepdrix

\[\Large V=x^2y \qquad\to\qquad\quad\qquad 208=x^2y\]\[\Large A=x^2+4xy \qquad\to\qquad 188=x^2+4xy\]

- zepdrix

So umm let's take our volume and write it in terms of x (solve for y).
Dividing by x^2 gives us,\[\Large \color{royalblue}{y=\frac{208}{x^2}}\]
We can plug this in for our y in the area equation.\[\Large 188=x^2+4x\color{royalblue}{y}\]

- zepdrix

\[\Large 188=x^2+4x\cdot \color{royalblue}{\frac{208}{x^2}}\]

- zepdrix

Understand what I did so far? :O
Think you can solve for x from this point? :)

- anonymous

yes!! I realized that my surface area formula was wrong!! so the 4xy is because it has 4 faces correct?

- zepdrix

ya :)

- anonymous

Thank you! i will check with you over the answer since I want to make sure i understand this :)

- zepdrix

k sounds good.

- anonymous

Ok, @zepdrix since i let my width =x i solved for x by plugging in the y we found into the surface area formula. I had to factor out X^2 and i was left with (1+832x)=187. i solved for x (my width) and got 187/832 or 0.2248. does that sound correct to you?

- zepdrix

\[\Large 188=x^2+4x\cdot \color{royalblue}{\frac{208}{x^2}}\]Factored out x^2? Hmm I don't think that's going to work :c\[\Large 188=x^2\left(1+x\cdot\frac{832}{x^4}\right)\]

- zepdrix

The x^2 is in the denominator of that term :o factoring x^2 out `takes away` another x^2.

- zepdrix

\[\Large 188=x^2+4x\cdot \color{royalblue}{\frac{208}{x^2}}\]I think what we want to do is, after simplifying this second term,\[\Large 188=x^2+\frac{832}{x}\]Multiply both sides by x and then solve the quadratic that comes out of that.

- zepdrix

Oh it's not a quadratic.. hmm woops

- anonymous

thank you! @zepdrix I'm trying to figure it out bit by bit with your help :)

- zepdrix

Hmm I must've made a boo boo somewhere.
Wolfram is saying there are no real solutions to that equation.
Grr one sec :d

- radar

Note that the base is rectangular, so the open top would also be rectangular (not square)

- zepdrix

Ah ty! There it is! :O

- zepdrix

|dw:1377985121301:dw|Two of these panels.

- zepdrix

|dw:1377985137702:dw|3 of these.

- anonymous

ooooh so my surface area is X^2+2xy??

- zepdrix

Hmm I think it works out to:
\[\Large A=2x^2+3xy\]

- zepdrix

We have 2 of the x^2 panels.
And 3 of the (xy) panels.

- anonymous

I see it! I'll re-use the old method and let you and @radar know what i get. Thank you both so much! :)

- zepdrix

\[\Large \color{royalblue}{y=\frac{208}{x^2}} \qquad\qquad 188=2x^2+3xy\]k :)

- zepdrix

Grrr I still can't get it to work out correctly >:O
@radar fix itttttttttttt :3

- radar

I'ma cipherin!. Neat problem

- anonymous

I don't understand this one bit >.<

- zepdrix

I was able to find lengths that work by graphing it:
https://www.desmos.com/calculator/xjp70ruffn
But I keep messing up the algebra for some reason..

- anonymous

so my length is 4 cm?? I've never dealt with graphs on this kind of problem..

- zepdrix

Ya graphing shouldn't be necessary for finding your x and y.
I'm probably just making a silly mistake somewhere.

- zepdrix

@satellite73

- anonymous

so far, i have 2X^2(1++312x)=188... this is my clearest step.. the rest is pretty messy.

- radar

|dw:1377986672911:dw||dw:1377987039548:dw|

- radar

Note, factoring was with help from Wolfphram :D

- radar

Messy is an apt description.

- zepdrix

hmm ya :(

- anonymous

@rada @zepdrix you guys are my heroes! I've been stuck for about 3 hours on this problem :/ I just have one question. In the beginning, did you factor out the x^2? the solution you provided makes perfect sense but i'm trying to find understand...

- radar

Y=13 when 4 is substituted in y=208/x^2

- radar

I placed the "slimmed" down cubic equation into Wolfphram graphing engine and it gave me the three roots with the (x-4) being the rational root. You can solve the other quadratic factor and look at those other 2 irrational roots.

- zepdrix

In the beginning? :o which step morn?

- anonymous

@radar Thank you so much! :) I'm not used to using wolfphram.. I'll definitely keep it in mind. I can't thank you or @zepdrix enough. This problem was tough!! :) i wish i could give you both medals :/ @zepdrix my question was cleared up since put the cleaned up version into wolfphram so no need to explain :) THANK YOU !!!

- zepdrix

Oh ok :3

- radar

I also use @zepdrix substitution for y. I got totally hung up sustituting for x

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