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Homework Help: Surface area of a open box

  1. Sep 8, 2010 #1
    1. The problem statement, all variables and given/known data

    Ok this is from a summer assignment that I had for calculus yet it's not calculus but it was graded by my calculus teacher so maybe there's something she knows that I don't from a calculus standpoint that allowed her to mark me wrong for this so here we go...

    Write a function for the surface area of an open box with a square base of side x and height y whose surface area is 100 square feet.

    Now strictly speaking I always thought that surface area was just that, area exposed to the surface of an object... I always thought that the assumption was that when not specified you just find the total area exposed to the surface, i.e. you don't subtract out the surface area that might be on the ground or what not because this is not a given, for example I remember like in other very simple math classes when we were asked to find the surface area of a sphere you would just use some formula... you didn't use calculus or anything to take out the area that is on the floor and is strictly speaking not exposed to the surface...

    so in this case I found the total surface area of a floating box just like you would a sphere when not told... even if you were to assume that it was on the ground with the given information you don't know which side of the box is on the floor? Is it on the floor with it's square base touching the floor or maybe one of the longer rectangular sides???? Or maybe it's on the floor and up against a wall in which case you would take out even more area that is not technically speaking exposed to the surface

    So am I justified in find the area of a floating OPEN box? I thought so as if she wanted us to assume it was on the floor she would clearly have to give us more information as I just explained... plus like in the 9th grade geometry when finding spheres you didn't take out the area that wasn't exposed to the surface...

    Well that was my logic...

    so anyways for my equation I got

    2x^2 + 8xy = 100 ft^2
    then I wrote it in equation form
    -x/4 + (25 ft^2)/2 = y

    and she marked it wrong I don't see how or why
    ok so the square base has a area of just x^2 but sense you got to count the bottom it's x^2 + x^2 or simply 2x^2 then there are 4 side panels of the box who have area xy but sense it's open you have to count it 8 times or simply 8xy then I set it equal to 100 ft^2 and then solved for y to make it a function

    and she marked it wrong why? Is my equation wrong or what?

    2. Relevant equations

    3. The attempt at a solution
  2. jcsd
  3. Sep 8, 2010 #2
    I still believe that I am justified no?
  4. Sep 8, 2010 #3


    Staff: Mentor

    The surface area is the area of the outer surface (the outside) of the box. "Area exposed to the surface" is, I suppose, a very convoluted way of saying surface area.
    It doesn't matter if it's floating or submerged or in the corner or whatever. You are justified in finding an expression for the surface area of an open box -- in fact, not only are you justified, that is what you are supposed to do.
    The (outer) surface of the box consists of a bottom (of area x2) and four sides (each of area xy).
    How does this follow from the preceding (and incorrect) equation? I have no idea what you did to get this equation.
  5. Sep 8, 2010 #4
    In the inside of the open box is a base with side x so the area would be x^2
    I thought of a floating box
    so therefore the base of the box on the underside which is also exposed to the surface also has a side of x and an area of x^2...
    let me know if that made no sense, so then I got 2x^2 for that part and used the same logic for the sides of the box with length y
  6. Sep 8, 2010 #5


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    You have a rectangular prism. Surface area of a rectangular prism is 2*(lenght*width+lenght*height+width*height)

    So 2*(x*y+x*z+y*z) = 100

    2xy + 2xz + 2yz = 100

    BUT since you have an open box, lets say the top or bottom or side is missing, you pick, I'm picking this version:

    xy + 2xz + 2yz = 100

    And yes x=y, so..
  7. Sep 8, 2010 #6
    Well see I was just told to find the surface area, outer or inner not specified, plus it doesn't matter as it is open right?
  8. Sep 8, 2010 #7
    Yes but it's open... it's not a solid rectangular prism with only 6 sides exposed to the surface in an open, floating box, there are ten sides exposed to the surface no?

    This is were I got into the whole thing well if it was on the ground there would only be nine exposed but it dosen't say and if she wanted it to be on the ground she would have to not only say so but say how it's sitting on the ground, is the base on the ground or a side, is it up against the wall etc.

    sense it was open I was like ok there are ten surfaces on the insdie of the box which I must include because it's open and there are also ten on the out, if it was floating, so I found the area using that logic
  9. Sep 8, 2010 #8


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    For the purpose of mathematics, we are going to pick one fluxable entity, I think..

    [PLAIN]http://www.k12math.com/math-concepts/algebra/surface_area/box_ex.png [Broken]

    I guess what I'm saying is.. you can't see all 10 sides of this box simultaneously.. who knows if they even exist.. they might be in a state of superposition.. I don't know. But what is certain is that if Im looking at the box I can 'see' the 6 sides to it even if there are really 12
    Last edited by a moderator: May 4, 2017
  10. Sep 8, 2010 #9
    Yes I know but there's no top it's open and if it were floating leaving a back, two ends, a front, and a bottom, now if it's floating the backsides of each of these pieces would also be exposed to the surface giving you a total of ten
  11. Sep 8, 2010 #10


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    Yes but the answer is 5 sides. You gotta love mathematics :biggrin:
  12. Sep 8, 2010 #11
    please explain why as i don't see why it's correct the box is open? Why did they tell me it was open if it didn't matter? please explain this to me I thought I was suppose to find the area of each pannal and then figuer it out with ten sides... please explain
  13. Sep 8, 2010 #12


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    Ok look the box is open, that means one side is missing, ok? Thats all.

    Look at the picture I've posted

    [PLAIN]http://www.k12math.com/math-concepts/algebra/surface_area/box_ex.png [Broken]

    Take away one side, now count, how many sides do you have? Use your fingers! 1.. 2.. 3.. 4.. 5! Good job :biggrin:

    If you were from the bottom, you would see the same exact sides, and there would be only 5 of them.

    You are now entering another dimension where you want to perceive all the sides and all dimensions simultaneously.. but wait, what is this.. you are infinitely small now!
    Last edited by a moderator: May 4, 2017
  14. Sep 8, 2010 #13


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    If you go to the cardboard store and buy a 1m by 1m piece of cardboard, they should charge you for 1 square meter of cardboard. If they charge you for 2 square meters because it has two sides, then you are being cheated. The conventional view of area of packing material includes the other side for free. That is really all there is to it. If you keep taking an eccentric view of what other people take for granted, it's going to be very hard for them to understand you and it's going to be very hard for you to solve problems because you are speaking a different language.
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