Question 935703
Intuition tells me this has a very expected symmetry to use.  The distance between any two opposite corners of the cube must be a diameter of the sphere.


That distance is found with the Pythagorean Theorem in THREE DIMENSIONS.  If assign s = length of edge of the cube,
{{{s^2+s^2+s^2=(2(4sqrt(3)))^2}}}.


{{{3s^2=4*16*3}}}
{{{s^2=4*16}}}
{{{s=2*4}}}
{{{highlight(s=8)}}}, the edge length of the cube.


The volume of this cube is {{{highlight(s^3=8^3=highlight(512))}}}.