Find the volume of the largest rectangular box with edges parallel to the axes that can be inscribed in the ellipsoid x^2/64+y^2/36+z^2/1=1

Question

Hint: By symmetry, you can restrict your attention to the first octant (where x,y,z≥0), and assume your volume has the form V=8xyz. Then arguing by symmetry, you need only look for points which achieve the maximum which lie in the first octant. Maximum volume:

Answer 1

here is a good place to start.
https://math.stackexchange.com/questions/531565/largest-box-fitting-inside-an-ellipsoid