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Homework Help: Optimization greatest possible volume Problem

  1. Oct 25, 2005 #1

    nrm

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    Question: [ A rectangular box, whose edges are parallel to the coordinate axes, is inscribed in the ellipsoid 96x^2 + 4y^2 + 4z^2 = 36, What is the greatest possible volume for such a box ]

    I realize that the volume of the box: V = (2x)(2y)(2z) = 8xyz
    Thus far I've solved for z^2 in the equation of the ellipsoid and then squared the volume so that I could make the substitution easier
    V^2 = 64(x^2)(y^2)(9-24x^2-y^2)
    Then I've taken the partial derivates of this to look cor critical points, but here I get an algebraic nightmare and can't find critical points. I'm wondering if my initial steps are correct, it's the only thing I could think of doing.

    Any help would be great. thank you
     
  2. jcsd
  3. Oct 25, 2005 #2
    what did you get for the partials
     
  4. Oct 25, 2005 #3

    nrm

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    Partial with respect to x:
    1152x(y^2)-6144(x^3)(y^2)-128x(y^4)

    y
    1152(x^2)y-3072(x^4)y-256(x^2)(y^3)
     
  5. Oct 26, 2005 #4

    HallsofIvy

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    Science Advisor

    The partial derivative of
    [tex]64(x^2)(y^2)(9-24x^2-y^2)[/tex]
    with respect to x is, by the product rule,
    [tex]128xy^2(9- 24x^2- y^2)- 3072x^3y[/tex]
    set that equal to 0 and you should be able to do a lot of cancelling.

    I would do this problem with "Lagrange multipliers" but you may not have had that yet.
     
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