document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=251474>Question 251474</A>: <I><SPAN STYLE=\"word-break: break-all;\"> We just blew some air into a spherical balloon and doubled its volume. By how much did we multiply the surface area ? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #183144 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=183144\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> formula for volume of a sphere is:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=v%5B1%5D+=+4%2F3+%2A+pi+%2A+r%5E3 ALIGN=MIDDLE  ALT=\"v%5B1%5D+=+4%2F3+%2A+pi+%2A+r%5E3\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "formula for surface area of a sphere is:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=s%5B1%5D+=+4+%2A+pi+%2A+r%5E2 ALIGN=MIDDLE  ALT=\"s%5B1%5D+=+4+%2A+pi+%2A+r%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "if we solve for r in the equation for the volume of the sphere, we get:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r%5E3+=+%283%2Av%29%2F%284%2Api%29 ALIGN=MIDDLE  ALT=\"r%5E3+=+%283%2Av%29%2F%284%2Api%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "take the cube root of both sides of this equation to get:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r+=+%28%283%2Av%29%2F%284%2Api%29%29%5E%281%2F3%29 ALIGN=MIDDLE  ALT=\"r+=+%28%283%2Av%29%2F%284%2Api%29%29%5E%281%2F3%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "if substitute for the value of r in the equation for the surface area of the sphere, then we get:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=s%5B1%5D+=+4+%2A+pi+%2A+r%5E2 ALIGN=MIDDLE  ALT=\"s%5B1%5D+=+4+%2A+pi+%2A+r%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > becomes:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=s%5B1%5D+=+4+%2A+pi+%2A+%28%28%283%2Av%29%2F%284%2Api%29%29%5E%281%2F3%29%29%5E2 ALIGN=MIDDLE  ALT=\"s%5B1%5D+=+4+%2A+pi+%2A+%28%28%283%2Av%29%2F%284%2Api%29%29%5E%281%2F3%29%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "if we double v, then this equation becomes:\r<BR>\n" );
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document.write( "if we divide s[2] by s[1] we get:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=s%5B2%5D%2Fs%5B1%5D+=+%284+%2A+pi+%2A+%28%28%286%2Av%29%2F%284%2Api%29%29%5E%281%2F3%29%29%5E2%29+%2F+%284+%2A+pi+%2A+%28%28%283%2Av%29%2F%284%2Api%29%29%5E%281%2F3%29%29%5E2%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "this simplifies to:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=s%5B2%5D%2Fs%5B1%5D+=+2%5E%282%2F3%29 ALIGN=MIDDLE  ALT=\"s%5B2%5D%2Fs%5B1%5D+=+2%5E%282%2F3%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > which is the same as:\r<BR>\n" );
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document.write( "multiply both sides of this equation by <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=s%5B1%5D ALIGN=MIDDLE  ALT=\"s%5B1%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" > and you get:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=s%5B2%5D+=+root%283%2C2%29%5E2%2As%5B1%5D ALIGN=MIDDLE  ALT=\"s%5B2%5D+=+root%283%2C2%29%5E2%2As%5B1%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "for example:\r<BR>\n" );
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document.write( "let the volume of the sphere be equal to 300.\r<BR>\n" );
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document.write( "we solve for the radius to get:\r<BR>\n" );
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document.write( "radius of the sphere equals 4.152830592\r<BR>\n" );
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document.write( "we confirm by substituting in the equation for the volume of the sphere to get:\r<BR>\n" );
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document.write( "v = 4/3 * pi * (4.152830592)^3 = 300 cubic units.\r<BR>\n" );
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document.write( "the surface area of the sphere is equal to 4 * pi * r^2 = 216.7196518 square units.\r<BR>\n" );
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document.write( "if the sphere doubles in volume, then the surface area of the new sphere should be equal to 216.7196518 * <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=root%283%2C2%29%5E2 ALIGN=MIDDLE  ALT=\"root%283%2C2%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > making the surface area of the new sphere equal to\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=s%5B2%5D+=+344.0210033 ALIGN=MIDDLE  ALT=\"s%5B2%5D+=+344.0210033\"   DISABLED_style= \"max-width:90%; height:auto;\" > square units\r<BR>\n" );
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document.write( "to confirm this is correct, we need to solve for the radius of the new sphere.\r<BR>\n" );
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document.write( "the volume of the new sphere is equal to 600 cubic units.\r<BR>\n" );
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document.write( "the radius of the new sphere is calculated to be 5.23223868 units.\r<BR>\n" );
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document.write( "the surface area of the new sphere is equal to 4 * pi * r^2 which becomes\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=s%5B2%5D+=+4+%2A+pi+%2A+%285.23223868%29%5E2+=+344.0210033 ALIGN=MIDDLE  ALT=\"s%5B2%5D+=+4+%2A+pi+%2A+%285.23223868%29%5E2+=+344.0210033\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "since this is the same as <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=216.7196518+%2A+root%283%2C2%29%5E2 ALIGN=MIDDLE  ALT=\"216.7196518+%2A+root%283%2C2%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > then we're good.\r<BR>\n" );
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document.write( "the answer to your question is:\r<BR>\n" );
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document.write( "the surface area was multiplied by <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=root%283%2C2%29%5E2 ALIGN=MIDDLE  ALT=\"root%283%2C2%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( " </SPAN>\n" );
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