\section*{Task 4 - 011014}

From a cubic piece of material (edge length \(a\)) the largest possible sphere is turned out. Which weighs more, the sphere or the swarf? Justify your answer!
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The sphere has a diameter \(a\) and thus a volume of \(\frac{\pi}{6} a^{3}\). The cube has a volume of \(a^{3}\), so the waste material has a volume of \(\left(1-\frac{\pi}{6}\right) a^{3}\). The volume of the sphere is therefore slightly larger than that of the waste, since it holds that: \(\pi>3 \Rightarrow \frac{\pi}{3}>1 \Rightarrow \frac{\pi}{6}>1-\frac{\pi}{6}\).