\section*{Problem 3 - 281013}

Given is a regular, five-sided, straight pyramid \(P\) with a height of \(h=10 \mathrm{~cm}\). By a planar cut parallel to the base, a regular, five-sided, and straight pyramid \(P^*\) is to be separated from this pyramid, whose volume \(V^*\) is one-third of the volume \(V\) of the original pyramid \(P\). How large is the height \(h^*\) of this separated pyramid \(P^*\)?

Hint: Estimate the expected result before the calculation! Will it lie

a) between \(2 \mathrm{~cm}\) and \(4 \mathrm{~cm}\),

b) between \(4 \mathrm{~cm}\) and \(6 \mathrm{~cm}\),

c) between \(6 \mathrm{~cm}\) and \(8 \mathrm{~cm}\),

d) between \(8 \mathrm{~cm}\) and \(9 \mathrm{~cm}\).
}

If \(S\) is the apex of both pyramids, then \(P^{*}\) is derived from \(P\) through central dilation. If \(k\) is the dilation factor, then for the volumes \(V^{*}=k^{3} V\). Therefore, the requirement \(V^{*}=\frac{1}{3} V\) is exactly met when \(k^{3}=\frac{1}{3}\), i.e., \(k=\frac{1}{\sqrt[3]{3}}\), and thus for the height, we have:

\[
h^{*}=k \cdot h=\frac{10}{\sqrt[3]{3}} \mathrm{~cm} \approx 6.9 \mathrm{~cm}
\]