Lagrange Multipliers, calc max volume of box

[hils0005]

Homework Statement

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

Homework Equations

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

The Attempt at a Solution

\nablaf=\nablag\lambda

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

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

4x^2\lambda=20y^2\lambda=10z^2\lambda

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

g=2(\sqrt{}5y)^2 +10y^2 + 5(\sqrt{}2y)^2=80
solving for y=\sqrt{}(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?

 

[Dick]

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.