document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=257194>Question 257194</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A cube and a sphere would have the same volume if the sphere’s radius were<BR>\n" );
document.write( "halved. What is the exact ratio of the sphere’s radius to the cube’s side? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #189117 by </B><A HREF=/tutors/aboutme.mpl?userid=Theo><B>Theo(13342)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Theo><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/cgi-bin/show-face.mpl?user=Theo><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=189117\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> if the cube and the sphere have the same volume, then the following equation is true.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=s%5E3+=+%284%2F3%29%2Api%2Ar%5E3 ALIGN=MIDDLE  ALT=\"s%5E3+=+%284%2F3%29%2Api%2Ar%5E3\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "solve for s to get:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=s+=+root%283%2C%28%284%2F3%29%2Api%2Ar%5E3%29%29 ALIGN=MIDDLE  ALT=\"s+=+root%283%2C%28%284%2F3%29%2Api%2Ar%5E3%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "this becomes:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=s+=+root%283%2C%28%284%2Api%29%2F3%29%29%2Aroot%283%2C%28r%5E3%29%29%29 ALIGN=MIDDLE  ALT=\"s+=+root%283%2C%28%284%2Api%29%2F3%29%29%2Aroot%283%2C%28r%5E3%29%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "which becomes:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=s+=+root%283%2C%28%284%2Api%29%2F3%29%29%2Ar ALIGN=MIDDLE  ALT=\"s+=+root%283%2C%28%284%2Api%29%2F3%29%29%2Ar\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "to confirm this is correct, we supply any value for r and calculate the volume of the sphere and the volume of the cube.\r<BR>\n" );
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document.write( "let r = 6.\r<BR>\n" );
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document.write( "volume of the sphere is equal to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%284%2F3%29%2Api%2A6%5E3 ALIGN=MIDDLE  ALT=\"%284%2F3%29%2Api%2A6%5E3\"   DISABLED_style= \"max-width:90%; height:auto;\" > = 904.7786842 cubic units.\r<BR>\n" );
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document.write( "side of the cube = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=root%283%2C%28%284%2Api%29%2F3%29%29%2A6 ALIGN=MIDDLE  ALT=\"root%283%2C%28%284%2Api%29%2F3%29%29%2A6\"   DISABLED_style= \"max-width:90%; height:auto;\" > = 9.671951724.\r<BR>\n" );
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document.write( "volume of the cube = (9.671951724)^3 = 904.7786842.\r<BR>\n" );
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document.write( "volumes are the same so the formula is good.\r<BR>\n" );
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document.write( "question was:\r<BR>\n" );
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document.write( "What is the exact ratio of the sphere’s radius to the cube’s side?\r<BR>\n" );
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document.write( "the original statement was?\r<BR>\n" );
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document.write( "A cube and a sphere would have the same volume if the sphere’s radius were<BR>\n" );
document.write( "halved.\r<BR>\n" );
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document.write( "since these cubes have the same volume, this means that the radius used was half the radius of the sphere.\r<BR>\n" );
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document.write( "this means that the sphere's radius is double whatever value we used to make the volumes equal.\r<BR>\n" );
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document.write( "in our formula for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=s+=+root%283%2C%28%284%2Api%29%2F3%29%29%2Ar ALIGN=MIDDLE  ALT=\"s+=+root%283%2C%28%284%2Api%29%2F3%29%29%2Ar\"   DISABLED_style= \"max-width:90%; height:auto;\" >, we'll replace \"r\" with \"r/2\" to get:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=s+=+root%283%2C%28%284%2Api%29%2F3%29%29%2A%28r%2F2%29 ALIGN=MIDDLE  ALT=\"s+=+root%283%2C%28%284%2Api%29%2F3%29%29%2A%28r%2F2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "this means that in our example, the radius of the sphere was actually 12 rather than 6.\r<BR>\n" );
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document.write( "plug that into the revised formula and we get:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=s+=+root%283%2C%28%284%2Api%29%2F3%29%29%2A%2812%2F2%29 ALIGN=MIDDLE  ALT=\"s+=+root%283%2C%28%284%2Api%29%2F3%29%29%2A%2812%2F2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > which becomes:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=s+=+root%283%2C%28%284%2Api%29%2F3%29%29%2A%286%29 ALIGN=MIDDLE  ALT=\"s+=+root%283%2C%28%284%2Api%29%2F3%29%29%2A%286%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > which is the same we had before when the volumes were equal.\r<BR>\n" );
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document.write( "so our formula is:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=s+=+root%283%2C%28%284%2Api%29%2F3%29%29%2A%28r%2F2%29 ALIGN=MIDDLE  ALT=\"s+=+root%283%2C%28%284%2Api%29%2F3%29%29%2A%28r%2F2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and we want to get the ratio of r to s.\r<BR>\n" );
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document.write( "in order to find that ratio, we want to get r/s = ???\r<BR>\n" );
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document.write( "we multiply both sides of our formula by 2 to get:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%2As+=+root%283%2C%28%284%2Api%29%2F3%29%29%2Ar ALIGN=MIDDLE  ALT=\"2%2As+=+root%283%2C%28%284%2Api%29%2F3%29%29%2Ar\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "we divide both sides of our formula by <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=root%283%2C%28%284%2Api%29%2F3%29%29 ALIGN=MIDDLE  ALT=\"root%283%2C%28%284%2Api%29%2F3%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%2As%2Froot%283%2C%28%284%2Api%29%2F3%29%29+=+r ALIGN=MIDDLE  ALT=\"2%2As%2Froot%283%2C%28%284%2Api%29%2F3%29%29+=+r\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "we divide both sides of this formula by s to get:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%2Froot%283%2C%28%284%2Api%29%2F3%29%29+=+r%2Fs ALIGN=MIDDLE  ALT=\"2%2Froot%283%2C%28%284%2Api%29%2F3%29%29+=+r%2Fs\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "our ratio is:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r%2Fs+=+2%2Froot%283%2C%28%284%2Api%29%2F3%29%29 ALIGN=MIDDLE  ALT=\"r%2Fs+=+2%2Froot%283%2C%28%284%2Api%29%2F3%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "we confirm this is true by taking any value for s and seeing if the formula holds.\r<BR>\n" );
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document.write( "we multiply both sides of this ratio by s to get:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r+=+%282%2As%29%2Froot%283%2C%28%284%2Api%29%2F3%29%29 ALIGN=MIDDLE  ALT=\"r+=+%282%2As%29%2Froot%283%2C%28%284%2Api%29%2F3%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "when we let s = 5, this formula becomes:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r+=+%2810%29%2Froot%283%2C%28%284%2Api%29%2F3%29%29 ALIGN=MIDDLE  ALT=\"r+=+%2810%29%2Froot%283%2C%28%284%2Api%29%2F3%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > = 6.203504909\r<BR>\n" );
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document.write( "we take half of this to get half of r = 3.101752454\r<BR>\n" );
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document.write( "volume of the cube is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5%5E3+=+125 ALIGN=MIDDLE  ALT=\"5%5E3+=+125\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "volume of the sphere when the radius is halved is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%284%2F3%29%2Api%2A%283.101752454%29%5E3+=+125 ALIGN=MIDDLE  ALT=\"%284%2F3%29%2Api%2A%283.101752454%29%5E3+=+125\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "we're good.\r<BR>\n" );
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document.write( "your answer is:\r<BR>\n" );
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document.write( "the ratio of the radius of the sphere to the side of the cube is:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r%2Fs+=+2%2Froot%283%2C%28%284%2Api%29%2F3%29%29 ALIGN=MIDDLE  ALT=\"r%2Fs+=+2%2Froot%283%2C%28%284%2Api%29%2F3%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "this is the ratio of the radius of the sphere before we halved it, as I believe that's what they are asking.\r<BR>\n" );
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document.write( " </SPAN>\n" );
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