document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=991842>Question 991842</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The diameter of a sphere is reduced by 10%. By what percent will the volume be reduced? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #611575 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=611575\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> since diameter = 2 times the radius, then if you reduce the diameter by 10%, you also reduce the radius by 10%.\r<BR>\n" );
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document.write( "for example:\r<BR>\n" );
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document.write( "assume the diameter is 10.<BR>\n" );
document.write( "reduce it by 10% and the diameter is now 9.\r<BR>\n" );
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document.write( "since the diameter is 10, the radius has to be 10/2 = 5.<BR>\n" );
document.write( "since the reduced diameter is 9, the reduced radius has to be 9/2 = 4.5.\r<BR>\n" );
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document.write( "5 - 4.5 is equal to .5 and .5 / 5 is equal to .10 which is equal to 10%, so the radius has been reduced by 10% as well.\r<BR>\n" );
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document.write( "the formula for the volume of a sphere is 4/3 * pi * r^3.\r<BR>\n" );
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document.write( "when you reduce the radius by 10%, the resulting radius is 90% of the original radius.\r<BR>\n" );
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document.write( "this is because r - .10 * r is equal to r * (1 - .10) which is equal to r * .9.\r<BR>\n" );
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document.write( "reduce the radius by 10% and the formula becomes v = 4/3 * pi * (.9*r)^3 which becomes v = 4/3 * pi * .9^3 * r^3 which then becomes v = 4/3 * pi * .729 * r^3.\r<BR>\n" );
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document.write( "this can be rewritten as v = .729 * 4/3 * pi * r^3.\r<BR>\n" );
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document.write( "the volume has been reduced by (1 - .729) = .271, or 27.1%.\r<BR>\n" );
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document.write( "let's take an example:\r<BR>\n" );
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document.write( "assume the diameter of the sphere is 24 inches.\r<BR>\n" );
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document.write( "the radius then has to be 12 inches because the length of the radius is half the length of the diameter.\r<BR>\n" );
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document.write( "the volume of the sphere is 4/3 * pi * 12^3 which is equal to 4/3 * pi * 1728.\r<BR>\n" );
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document.write( "reduce the diameter by 10%.\r<BR>\n" );
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document.write( "24 - .10 * 24 = 24 - 2.4 = 21.6\r<BR>\n" );
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document.write( "this is the same as .9 * 24 which is equal to 21.6.\r<BR>\n" );
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document.write( "this is because 24 - .10 * 24 is equal to 24 * (1 - .10) which is equal to .9 * 24.\r<BR>\n" );
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document.write( "the radius of this sphere is equal to 21.6 / 2 = 10.8\r<BR>\n" );
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document.write( "the volume of this sphere is equal to 4/3 * pi * (10.8)^3 which is equal to 4/3 * pi * 1259.712.\r<BR>\n" );
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document.write( "the volume of the original sphere is 4/3 * pi * 1728.\r<BR>\n" );
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document.write( "the volume of the reduce sphere is 4/3 * pi * 1259.712.\r<BR>\n" );
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document.write( "the difference in the volume is 4/3 * pi * 1728 - 4/3 * pi * 1259.712.\r<BR>\n" );
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document.write( "this can be rewritten as 4/3 * pi * (1728 - 1259.712) which can then be simplified to 4/3 * pi * 468.288.\r<BR>\n" );
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document.write( "the ratio of this difference to the original volume is:\r<BR>\n" );
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document.write( "(4/3 * pi * 468.288) / (4/3 * pi * 1728).\r<BR>\n" );
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document.write( "4/3 * pi / 4/3 * pi cancels out because it is equal to 1 and you are left with:\r<BR>\n" );
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document.write( "the ratio of the difference is equal to 468.288 / 1728 which is equal to .271 which is equal to 27.1%.\r<BR>\n" );
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document.write( "reduce the diameter of the sphere by 10% and you reduce the volume of the sphere by 27.1%.\r<BR>\n" );
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document.write( " </SPAN>\n" );
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