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Lagrange Multipliers, calc max volume of box

  1. Nov 18, 2008 #1
    1. The problem statement, all variables and given/known data

    Point P(x,y,z) lies on the part of the ellipsoid 2x^2 + 10y^2 + 5z^2 = 80 that is in the first octant of space. It is also a vertex of a rectangular parallelpiped each of whose sides are parallel to a coordinate plane. Use Method of LaGrange Multipliers to determine the coordinates of P so that the box has a max volume and calculate the max



    2. Relevant equations
    f(x,y,z)=xyz g(x,y,z)=2x^2+10y^2+5z^2=80


    3. The attempt at a solution

    [tex]\nabla[/tex]f=[tex]\nabla[/tex]g[tex]\lambda[/tex]

    1.yz=4x[tex]\lambda[/tex]
    2.xz=20y[tex]\lambda[/tex]
    3.xy=10z[tex]\lambda[/tex]

    I multiplied equation 1 by x, 2 by y and 3 by z

    4x^2[tex]\lambda[/tex]=20y^2[tex]\lambda[/tex]=10z^2[tex]\lambda[/tex]

    I then put x and z in terms of y and put into constraint
    4x^2=20y^2 10z^2=20y^2
    x=[tex]\sqrt{}[/tex]5 y z=[tex]\sqrt{}[/tex]2 y

    g=2([tex]\sqrt{}[/tex]5y)^2 +10y^2 + 5([tex]\sqrt{}[/tex]2y)^2=80
    solving for y=[tex]\sqrt{}[/tex](8/3)

    I'm not sure if I'm on the right track or if this is way off, if correct do I just do the same proceedure to find x and z?
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
  3. Nov 19, 2008 #2

    Dick

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    Science Advisor
    Homework Helper

    You are doing fine. You don't need to repeat the procedure for x and z. You already have x=sqrt(5)*y and z=sqrt(2)*y. Once you've got y, you've got everything.
     
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