# Maximize Volume of a Rectangular Box

by dtl42
Tags: maximize, rectangular, volume
 Math Emeritus Sci Advisor Thanks PF Gold P: 39,682 One thing students seem to have trouble realizing is that applications don't typically come with coordinate systems attached! You have a sphere, sittin there in space, with a rectangle inscribed in it. You can't write equations until you have set up a coordinate system. The obvious thing, I think, is to choose your coordinate system so that (0,0,0) is at the center of the sphere and then the equation of the sphere is $x^2+ y^2+ z^2= 1$. That still leaves the orientation of the axes- again, it would strike me as simplest to choose the axes parallel to the edges of the box. Now, one corner of the box will be in the first octant, (x, y, z) with x, y, and z positive, and, of course, $x^2+ y^2+ z^2= 1$. It should be easy, using the fact that the edges of the are parallel to the axes, and using the symmetry of the sphere, to write down the coordinates of the other 7 corners and so find the lengths of the edges and the volume as a function of x, y, and z.