In a right circular cone, the semi-vertical angle of which is θ, a cube is placed so that four of its vertices are on the base and four on the curved surface. Prove that as θ varies the maximum of the ratio for the volume of the cube to the volume of the cone occurs when sin θ = 1/3.

Note: The semi-vertical angle of a cone is half the vertex angle.
For a cone with height h and radius r, tan θ = r/h.