A right prism has a base that is an equilateral triangle. The height of the prism is equal to the height of the base. If the volume of the prism is 81, what are the lengths of the sides of the base?
+14903 A right prism has a base that is an equilateral triangle. The height of the prism is equal to the height of the base. If the volume of the prism is 81, what are the lengths of the sides of the base?
Ein rechtes Prisma hat eine Basis, die ein gleichseitiges Dreieck ist. Die Höhe des Prismas entspricht der Höhe der Basis. Wenn das Volumen des Prismas 81 beträgt, wie lang sind die Seiten der Basis?
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Equilateral triangle ABC
\(h=\sqrt{a^2-(\frac{a}{2})^2}\)
\(V_{prism}=(\frac{ah}{2})h=\frac{ah^2}{2}=81\\ a(a^2- (\frac{a}{2})^2)=81\cdot 2\\ a^3-\frac{a^3}{4}=162\\ 4a^3-a^3=4\cdot 162\)
\(3a^3=648\)
\(a^3=216\)
\(a=6\)
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