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 =  2.476 cm

Solution:

r=10 cm h=12 cm  V1=π r2 h=3.1416 102 123769.9112 cm3 V2=V1/2=3769.9112/21884.9556 cm3  r2:h2=r:h V2=π r22 h2 V2=π (r/h h2)2 h2  h2=V2 h2π r23=1884.9556 1223.1416 10239.5244 cm  x=hh2=129.52442.47562.476 cm   Correctness test:  r2=r/h h2=10/12 9.52447.937 cm V22=π r22 h2=3.1416 7.9372 9.52441884.9556 cm3r=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=3769.9112/2 \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/h \cdot \ h_{2})^2 \cdot \ h_{2} \ \\ \ \\ h_{2}=\sqrt[3]{ \dfrac{ V_{2} \cdot \ h^2 }{ \pi \cdot \ r^2 } }=\sqrt[3]{ \dfrac{ 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 \doteq 2.476 \ \text{cm} \ \\ \ \\ \text{ Correctness test: } \ \\ 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



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