document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1151079>Question 1151079</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A hemisphere and a right circular cone on equal bases are of equal height. find the ratio of their volumes. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #772693 by </B><A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><B>jim_thompson5910(35256)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=772693\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font face=\"times\" color=\"black\" size=\"3\"><BR>\n" );
document.write( "Vs = Volume of sphere<BR>\n" );
document.write( "Vs = (4/3)*pi*r^3\r<BR>\n" );
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document.write( "A hemisphere is half a sphere.<BR>\n" );
document.write( "Vh = Volume of hemisphere<BR>\n" );
document.write( "Vh = (1/2)*Vs<BR>\n" );
document.write( "Vh = (1/2)*(4/3)*pi*r^3<BR>\n" );
document.write( "Vh = (4/6)*pi*r^3<BR>\n" );
document.write( "Vh = (2/3)*pi*r^3\r<BR>\n" );
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document.write( "We're told that both the hemisphere and cone have the same height. <BR>\n" );
document.write( "The height of the hemisphere is the radius r, so for the cone, h = r.<BR>\n" );
document.write( "Vc = Volume of cone<BR>\n" );
document.write( "Vc = (1/3)*pi*r^2*h<BR>\n" );
document.write( "Vc = (1/3)*pi*r^2*r ... plug in h = r<BR>\n" );
document.write( "Vc = (1/3)*pi*r^3\r<BR>\n" );
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document.write( "-------------------------------------------\r<BR>\n" );
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document.write( "Divide the hemisphere volume over the cone volume.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=Vh%2FVc+=+%28%282%2F3%29%2Api%2Ar%5E3%29%2F%28%281%2F3%29%2Api%2Ar%5E3%29 ALIGN=MIDDLE  ALT=\"Vh%2FVc+=+%28%282%2F3%29%2Api%2Ar%5E3%29%2F%28%281%2F3%29%2Api%2Ar%5E3%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=Vh%2FVc+=+%28%282%2F3%29%2Across%28pi%29%2Ar%5E3%29%2F%28%281%2F3%29%2Across%28pi%29%2Ar%5E3%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > The pi terms cancel.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=Vh%2FVc+=+%28%282%2F3%29%2Ar%5E3%29%2F%28%281%2F3%29%2Ar%5E3%29 ALIGN=MIDDLE  ALT=\"Vh%2FVc+=+%28%282%2F3%29%2Ar%5E3%29%2F%28%281%2F3%29%2Ar%5E3%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=Vh%2FVc+=+%28%282%2F3%29%2Across%28r%5E3%29%29%2F%28%281%2F3%29%2Across%28r%5E3%29%29 ALIGN=MIDDLE  ALT=\"Vh%2FVc+=+%28%282%2F3%29%2Across%28r%5E3%29%29%2F%28%281%2F3%29%2Across%28r%5E3%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > The r^3 terms cancel.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=Vh%2FVc+=+%282%2F3%29%2F%281%2F3%29 ALIGN=MIDDLE  ALT=\"Vh%2FVc+=+%282%2F3%29%2F%281%2F3%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=Vh%2FVc+=+%282%2F3%29%2A%283%2F1%29 ALIGN=MIDDLE  ALT=\"Vh%2FVc+=+%282%2F3%29%2A%283%2F1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Flip the second fraction and multiply.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=Vh%2FVc+=+%282%2A3%29%2F%283%2A1%29 ALIGN=MIDDLE  ALT=\"Vh%2FVc+=+%282%2A3%29%2F%283%2A1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=Vh%2FVc+=+6%2F3 ALIGN=MIDDLE  ALT=\"Vh%2FVc+=+6%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=Vh%2FVc+=+2 ALIGN=MIDDLE  ALT=\"Vh%2FVc+=+2\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<font color=red size=4>The ratio of the hemisphere volume to the cone volume is 2:1. </font>\r<BR>\n" );
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document.write( "This means the hemisphere has twice the volume of the cone.<BR>\n" );
document.write( "Put another way,<BR>\n" );
document.write( "Vh = 2*Vc<BR>\n" );
document.write( "which can be rearranged to<BR>\n" );
document.write( "Vc = (1/2)*Vh\r<BR>\n" );
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document.write( "This only works if the cone and hemisphere share the same circular base, and also have the same height (h = r).\r<BR>\n" );
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document.write( "--------------------\r<BR>\n" );
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document.write( "side note:\r<BR>\n" );
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document.write( "If we start with Vh = 2*Vc and plug in Vc = (1/3)*pi*r^3, then we get,<BR>\n" );
document.write( "Vh = 2*Vc<BR>\n" );
document.write( "Vh = 2*(1/3)*pi*r^3<BR>\n" );
document.write( "Vh = (2/3)*pi*r^3<BR>\n" );
document.write( "Or we could start with Vc = (1/2)*Vh and plug in Vh = (2/3)*pi*r^3 to get,<BR>\n" );
document.write( "Vc = (1/2)*Vh<BR>\n" );
document.write( "Vc = (1/2)*(2/3)*pi*r^3<BR>\n" );
document.write( "Vc = (1/3)*pi*r^3<BR>\n" );
document.write( "This helps confirm our answer.<BR>\n" );
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