A fountain mounted on a pole consists of a marble sphere resting in a bronze bowl. The marble sphere touches the four inner walls of the bronze bowl at exactly one point each. The bronze bowl is described in the model by the lateral faces of the pyramid ABCDS, the marble sphere by a sphere with center $M ( 0 | 0 | 4 )$ and radius $r$. The $x _ { 1 } x _ { 2 }$-plane of the coordinate system represents the horizontally running ground in the model; one unit of length corresponds to one decimeter in reality. Determine the diameter of the marble sphere to the nearest centimeter.
A fountain mounted on a pole consists of a marble sphere resting in a bronze bowl. The marble sphere touches the four inner walls of the bronze bowl at exactly one point each. The bronze bowl is described in the model by the lateral faces of the pyramid ABCDS, the marble sphere by a sphere with center $M ( 0 | 0 | 4 )$ and radius $r$. The $x _ { 1 } x _ { 2 }$-plane of the coordinate system represents the horizontally running ground in the model; one unit of length corresponds to one decimeter in reality.
Determine the diameter of the marble sphere to the nearest centimeter.