Basic questions about finding a volume formula for a box inside of an ellipsoid,
I want to maximize the volume of a box, with sides parallel to the xy, xz and yz-planes, with the box inside of an ellipsoid
So, my answer, using Lagrange multipliers, turns out to be wrong, at least comparing my work to the solution given.
I had thought to maximize "length times height times width", so I figured that the objective function should be .
The solution instead maximizes a different function:
I am guessing that the solution is indeed correct? And the point I probably missed was this: the ellipsoid is centered at 0. So, sketching out the box on paper, "length times height times width" does look like .
What do you think?
Also, how does an equation of an ellipsoid not centered at 0 look like? Would it be something like this:
would this be an ellipsoid centered at (1,2,3)?