\section*{Problem 1 - 111221}

Given are two cubes with edge lengths \(a\) and \(b\), respectively.

We are looking for a right prism with a square base, whose volume is equal to the sum of the volumes of the cubes and whose height is equal to the sum of the lengths of the cube edges.

a) Calculate the side length \(c\) of the square base of such a prism.

b) Provide a construction for a segment of the length \(c\) determined in a).
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a) For the volume \(V\) of the prism, we have \(V=c^{2} h\) with \(h=a+b\). From \(a^{3}+b^{3}=V=c^{2}(a+b)\), we obtain

\[
c^{2}=\frac{a^{3}+b^{3}}{a+b}=a^{2}-a b+b^{2}
\]

b) From the equation \(c^{2}=a^{2}-a b+b^{2}=a^{2}+b^{2}-2 a b \cos \left(60^{\circ}\right)\), we can derive a construction rule using the cosine law: Construct a triangle from \(a, b\) with the included angle \(\gamma=60^{\circ}\). The third side of the triangle will then have the desired length \(c\).

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