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

$$\nabla$$f=$$\nabla$$g$$\lambda$$

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.