Given a sphere of fixed radius a. A right circular cone is is to be
found which will enclose the sphere such that the sphere is tangent
to the cone's lateral surface, the sphere is tangent to the cone's
base at its center, and the ratio Ac/As is minimised where Ac is the
cone's surface area (both lateral and base) and As the sphere's
surface area.

To keep solutions uniform let's denote the cone's altitude by h, base
radius by r and slant height by L.