$Floating$ $Ice$ $Cube$

A cube of ice of edge $4$ $cm$ is placed in an empty cylindrical tumbler of inner diameter $6$ $cm$ . Assume that ice melts uniformly from each side so that it always retains its cubical shape. Find the length of the edge of the ice cube $(in$ $centimeters)$ at the instant it just leaves contact with the bottom of the glass.

$Details$ $and$ $Assumptions$

$Density$ $of$ $Ice = 900 kg/m^3$ $;$ $Density$ $of$ $Water = 1000 kg/m^3$