An Ice Problem

Preparing to mix a negroni, a bartender places a sphere of ice with a radius of 2 cm into an empty martini glass. The cone-shaped part of the glass has a height of 6 cm and a base angle of 60°. Assuming the ice sphere would be completely melted in 1 hour, how long would it take for the bottom of the ice sphere to touch the surface of the melt-water accumulating beneath it?

HINT: Don't forget that the ice sphere will sink lower in the glass as it melts.