Find the maximum volume of a box with faces parallel to thexy, xz, yz-planes that lies entirely inside the region enclosed by the cones
z = sqrt(x^2 + 2y^2) - 2 and
z = 2 - sqrt(x^2 + 2y^2)
The enclosed region is symmetric about xy-plane.
So, if z=k is on of the plane, then the other is z=-k.
If we cut the enclosed area by the plane z=k, we will get an ellipse
or,
The area of the rectangle having largest area(with sides parallel to ellipse axes) is
Hence, the volume of box is
For maximum volume, value of k is found to be
What method are you supposed to use ?
It looks like a Lagrange Multiplier problem to me.
Choose a point say on the cone
in the first octant, (so are all positive ), and form your box based on this point. It will have a volume of .
Now let
and differentiate partially wrt and .
Put each partial derivative equal to zero and solve the resulting equations simultaneously.
I get the values ,
in which case the maximum volume is