2025 NYJC P1 Q2

An ornament consists of two identical solid right circular cones whose flat bases are separated by a solid sphere, such that the sphere is in contact with the two bases. The axis passes through the vertex of each cone and the centre of the sphere. It is given that the distance between the vertices of the cones is \(2l\), and the radii of the bases of the cones are \(\sqrt{\frac{3}{2}}l\), where \(l\) is a constant. Find the radius of the sphere in terms of \(l\) if the total volume of the ornament is to be a minimum.

[The volume of a sphere with radius \(r\) is given by \(V = \frac{4}{3}\pi r^3\).][6]