Cutting cone

A cone with a base radius of 10 cm and a height of 12 cm is given. At what height above the base should we divide it by a section parallel to the base so that the volumes of the two resulting bodies are the same? Express the result in cm.

Correct result:

x =  2.4756 cm

Solution:

$$\begin{gathered} r=10\text{ cm } \\ h=12\text{ cm } \\ {V}_1=\pi \cdot r^2\cdot h=3.1416\cdot 10^2\cdot 12\doteq 3769.9112{\text{cm}}^3 \\ {V}_2=V_1\mathrm{/}2=3769.9112\mathrm{/}2\doteq 1884.9556{\text{cm}}^3 \\ {r}_2:h_2=r:h \\ {V}_2=\pi \cdot r_2^2\cdot h_2 \\ {V}_2=\pi \cdot (r\mathrm{/}h\cdot h_2{)}^2\cdot h_2 \\ {h}_2=\sqrt[3]{{\displaystyle \frac{V_2\cdot h^2}{\pi \cdot r^2}}}=\sqrt[3]{{\displaystyle \frac{1884.9556\cdot 12^2}{3.1416\cdot 10^2}}}\doteq 9.5244\text{ cm } \\ x=h-h_2=12-9.5244\doteq 2.4756=2.4756\text{ cm } \\ \text{ Verifying Solution: } \\ {r}_2=r\mathrm{/}h\cdot h_2=10\mathrm{/}12\cdot 9.5244\doteq 7.937\text{ cm } \\ {V}_{22}=\pi \cdot r_2^2\cdot h_2=3.1416\cdot 7.937^2\cdot 9.5244\doteq 1884.9556{\text{cm}}^3 \end{gathered}$$

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