document.write( "Question 1029812: How far from the vertex of a right circular cone must a plane be passed in order to divide the volume into two equal parts? three equal parts? \n" ); document.write( "

Algebra.Com's Answer #645021 by KMST(5328) $\"\"$ $\"About$

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It is not clear to me how are you going to angle that plane you are using to slice the cone into two parts. Parallel to the base of the cone?
\n" ); document.write( "Also, are you intending to use two parallel planes to slice that cone into three equal parts?\r
\n" ); document.write( "\n" ); document.write( "A plane is parallel to the base of the right circular cone of height $\"H\"$ ,
\n" ); document.write( "at a distance $\"h\"$ from the vertex,
\n" ); document.write( "it would delimit a smaller cone similar to the original cone.
\n" ); document.write( "The ratio of their heights is $\"H%2Fh\"$ .
\n" ); document.write( "The ratio of the volume of the original, larger cone, $\"V\"$ , $\"v\"$ ,
\n" ); document.write( "to the volume of the smaller cone,
\n" ); document.write( "is $\"V%2Fv=%28H%2Fh%29%5E3\"$ .
\n" ); document.write( "
\n" ); document.write( "If we want the plane to divide the volume into two equal parts,
\n" ); document.write( "we want $\"v=V-v\"$ <---> $\"2v=V\"$ <---> $\"V%2Fv=2\"$ .
\n" ); document.write( "For that we need
\n" ); document.write( " $\"%28H%2Fh%29%5E3=2\"$ --> $\"H%2Fh=root%283%2C2%29\"$ --> $\"h=H%2A%281%2Froot%283%2C2%29%29\"$ .
\n" ); document.write( "For a more elegant expression, $\"h=H%2A%28root%283%2C4%29%2F2%29\"$ .
\n" ); document.write( "For an approximate solution, $\"h=0.7937H\"$ .
\n" ); document.write( "To divide the volume into two equal parts, a plane parallel to the base of a right circular cone should pass at a distance from the vertex of about $\"0.7937\"$ times the height of the cone.
\n" ); document.write( "
\n" ); document.write( "If you use two planes parallel to the base of a cone,
\n" ); document.write( "at distances $\"h%5B1%5D\"$ and $\"h%5B2%5D\"$ from the vertex,
\n" ); document.write( "with $\"h%5B1%5D%3Ch%5B2%5D\"$
\n" ); document.write( "to slice that cone into three parts of equal volume,
\n" ); document.write( "The volume, $\"v%5B1%5D\"$ , of the small cone with its base $\"h%5B1%5D\"$ from the vertex will be
\n" ); document.write( " $\"v%5B1%5D=%281%2F3%29V\"$ .
\n" ); document.write( "Since , $\"v%5B1%5D%2FV=%28h%5B1%5D%2FH%29%5E3\"$ , then
\n" ); document.write( " $\"%28h%5B1%5D%2FH%29%5E3=1%2F3\"$ --> $\"h%5B1%5D%2FH=root%283%2C1%2F3%29=1%2Froot%283%2C3%29=root%283%2C9%29%2F3\"$ .
\n" ); document.write( "The approximate value is $\"h%5B1%5D%2FH=0.6934\"$ (to four decimal places).
\n" ); document.write( "So, to lop off the top

times the height of the cone.
\n" ); document.write( "The cone you would get slicing only at a distance $\"h%5B2%5D\"$ ,
\n" ); document.write( "would have a volume twice as large as the cone above $\"h%5B1%5D\"$ from the vertex:
\n" ); document.write( " $\"v%5B2%5D=2v%5B1%5D\"$ <--> $\"v%5B2%5D%2Fv%5B1%5D=2\"$ , with
\n" ); document.write( " $\"%28h%5B2%5D%2Fh%5B1%5D%29%5E3=2\"$ <--> $\"h%5B2%5D%2Fh%5B1%5D=root%283%2C2%29\"$ <--> $\"h%5B2%5D=h%5B1%5D%2Aroot%283%2C2%29\"$ .
\n" ); document.write( "Since we had $\"h%5B1%5D%2FH=root%283%2C9%29%2F3\"$ <--->. $\"h%5B1%5D=H%2Aroot%283%2C9%29%2F3\"$ ,
\n" ); document.write( " $\"h%5B2%5D=H%2Aroot%283%2C2%29%2Aroot%283%2C9%29%2F3=H%2Aroot%283%2C18%29%2F3\"$ .
\n" ); document.write( "For an approximate value, $\"h%5B2%5D=0.8736H\"$ .
\n" ); document.write( "So, to slice a cone into three equal volume parts using two slicing planes parallel to the base,
\n" ); document.write( "cut at distances from the vertex of $\"0.6394\"$ times the cone's height and $\"0.8736\"$ times the cone's height. \n" ); document.write( "

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