# Hemispherical Duck Pond

Calculus Level 3

A duck pond was built with a plastic liner in the shape of a hemisphere, radius $$R$$. When it rains in winter the pond fills up to the brim, so it is $$R$$ deep in the middle.

During the summer the water evaporates, so it is only half as deep $$\frac R2$$ in the middle.

What fraction of the full volume does the pond hold when it evaporates to half the depth in summer?

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