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 answer:

x =  2.4756 cm

Step-by-step explanation:

$$\begin{gathered} r=10\text{cm} \\ h=12\text{cm} \\ {V}_1=\pi \cdot r^2\cdot h=3.1416\cdot 10^2\cdot 12\doteq 3769.9112{\text{cm}}^3 \\ {V}_2=V_1/2=\frac{1146053}{304}/2=\frac{1146053}{304\cdot 2}=\frac{1146053}{608}\doteq 1884.9556{\text{cm}}^3 \\ {r}_2:h_2=r:h \\ {V}_2=\pi \cdot r_2^2\cdot h_2 \\ {V}_2=\pi \cdot (r/h\cdot h_2{)}^2\cdot h_2 \\ {h}_2=\sqrt[3]{\frac{V_2\cdot h^2}{\pi \cdot r^2}}=\sqrt[3]{\frac{1884.9556\cdot 12^2}{3.1416\cdot 10^2}}\doteq 9.5244\text{cm} \\ x=h-h_2=12-9.5244\doteq 2.4756\text{cm} \\ \text{ Verifying Solution: } \\ {r}_2=r/h\cdot h_2=10/12\cdot 9.5244\doteq 7.937\text{cm} \\ {V}_{22}=\pi \cdot r_2^2\cdot h_2=3.1416\cdot 7.937^2\cdot 9.5244\doteq 1884.9556{\text{cm}}^3 \end{gathered}$$

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