A right circular cone sits on a table, pointing up. The cross-section triangle, perpendicular to the base, has a vertex angle of 60 degrees. The diameter of the cone's base is \(12\sqrt3\) inches. A sphere is placed inside the cone so that it is tangent to the sides of the cone and sits on the table. What is the volume, in cubic inches, of the sphere? Express your answer in terms of \(\pi\).

Guest Apr 19, 2021