Find the maximum volume of a rectangular box that is inscribed in a sphere of radius r.?

Let's start with a cube with a side of 1. Pythagorus showed that the diagonal must be √(x²+y²+z²) = √(1²+1²+1²) ≈ √3. So, if the radius is R, then the length of half of the diagonal is R, and the length of the diagonal is 2R, and the length of a side is 2R/√3. The volume is thus (2R/√3)³, which still needs to be put into simplest terms.