(adsbygoogle = window.adsbygoogle || []).push({}); 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

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# Homework Help: Lagrange Multipliers, calc max volume of box

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