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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