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Problem of finding the volume of a box.

  1. Oct 9, 2011 #1
    1. The problem statement, all variables and given/known data

    The airlines accept a box if lenght + width + height <= 62. If h is fixed, show that the maximum volume (62-w-h)wh is V = h(31 - h/2)^2.

    3. The attempt at a solution

    Well, I multipied:

    V(h) = (62-w-h)wh
    V(h) = 62wh - w^2h - wh^2

    dv/dt = 62w - w^2 - 2w = 60w - w^2

    0 = 60w - w^2
    w = 60.

    But this can not be. This is wrong. I tried other things but all of my attemps are wrong...
     
  2. jcsd
  3. Oct 9, 2011 #2

    HallsofIvy

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    What you have done makes no sense at all. You write "V(h)" when V depends on both w andh as variables. And then you write "dv/dt" when there is NO "t" variable at all!

    A differentiable function of two variables will have max or min only where its gradient is 0 or, equivalently, [itex]\partial V/\partial w= 0[/itex] and [itex]\partial V/\partial h= 0[/itex].
     
  4. Oct 9, 2011 #3
    Thank you for your answer, HallsofIvy.

    "dv/dt" well, yes. That was a typo from my part. My bad. Clearly volume does not depent on t...

    I see that you have written partial derivative notation. I am assuming that your suggestion is to use partials, since volume seems to depend on two variables, namely h and w. But this exercise is in a problem set of a single-variable calculus textbook. I guess this can be solved that way, but for the moment I think the textbook expects me to solve the problem without the aid of partials. (I don't know know anything of partials. I only know something about its notation.)

    My thought is that since V = HLW, I would get something of the form V = h times M^2, if I assume L = W = M^2, and then substitute. But I don't know how to prove this assumption and I have the feeling that it is wrong...
     
    Last edited: Oct 9, 2011
  5. Oct 9, 2011 #4
    But the problem statement says h is fixed. I am not sure what does the textbook means with "fixed" (I am not an english native speaker.) I suppose it means that h is not a variable, since it is "fixed"; it does not change. If that is the case, then V depends on L and W. But the problem ask me to show that (62-w-h)wh is V = h(31 - h/2)^2. Here "V" seems to be in function of h; a some sort of contradiction. I am a little bit confused...
     
  6. Oct 10, 2011 #5
    Well, certainly I was confused yesterday. Happily now I undestand the problem:

    V = (62 - h - w)wh,

    V = 64wh - h^2w - w^2h.

    dv/dw = 62h - h^2 - 2wh,

    0 = 62h - h^2 - 2wh,

    w = 62h - h^2/ 2h

    w = 31 - h/2.

    It turns out that L = W, because L = 62 - h - w, or L = 62 - h - (31 - h/2), or L = 31 - h/2.

    Then, if we let W^2 = LW, it is clear that V = W^2H. Finally we have (31 - h/2)^2 times h.

    Thanks! See you next time!
     
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