document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=92665>Question 92665</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Senai customized her bicycle by exchanging the front wheel for a wheel that had one half the diameter of the back wheel. Now when Senai rides the bicycle, how many revolutions does the front wheel make for each revolution of the back wheel? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #67476 by </B><A HREF=/tutors/aboutme.mpl?userid=bucky><B>bucky(2189)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=bucky><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=67476\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> When the rear wheel makes 1 rotation, the bicycle moves forward a distance equal to the <BR>\n" );
document.write( "circumference of the rear wheel. The circumference of the rear wheel is given by the equation:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=C%5Br%5D+=+pi+%2A+D ALIGN=MIDDLE  ALT=\"C%5Br%5D+=+pi+%2A+D\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "where <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=C%5Br%5D ALIGN=MIDDLE  ALT=\"C%5Br%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" > is the circumference of the rear wheel and D is the diameter of the rear wheel.<BR>\n" );
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document.write( "When the bicycle moves forward an amount equal to the circumference of the rear wheel, the smaller <BR>\n" );
document.write( "front wheel must rotate enough times so that its circumference travels the same amount of <BR>\n" );
document.write( "distance as the circumference of the rear wheel.<BR>\n" );
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document.write( "The circumference of the front wheel is given by the equation:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=C%5Bf%5D+=+pi+%2A+d ALIGN=MIDDLE  ALT=\"C%5Bf%5D+=+pi+%2A+d\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "where d is the diameter of the front wheel and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=C%5Bf%5D ALIGN=MIDDLE  ALT=\"C%5Bf%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" > is the circumference of the front <BR>\n" );
document.write( "wheel. But the problem tells you that to customize the bicycle the new front wheel <BR>\n" );
document.write( "has a diameter equal to half the diameter of the rear wheel or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=d+=+D%2F2 ALIGN=MIDDLE  ALT=\"d+=+D%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" >. So in the <BR>\n" );
document.write( "equation for the circumference of the front wheel we can replace d by <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=D%2F2 ALIGN=MIDDLE  ALT=\"D%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" >. When that is <BR>\n" );
document.write( "done the equation for the circumference of the front wheel becomes:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=C%5Bf%5D+=+pi+%2A+D%2F2 ALIGN=MIDDLE  ALT=\"C%5Bf%5D+=+pi+%2A+D%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Now the problem is translated to \"When the back wheel rotates once the bicycle travels forward<BR>\n" );
document.write( "a distance of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=C%5Br%5D+=+pi+%2A+D ALIGN=MIDDLE  ALT=\"C%5Br%5D+=+pi+%2A+D\"   DISABLED_style= \"max-width:90%; height:auto;\" >. How many times does the front wheel have to rotate so<BR>\n" );
document.write( "that the distance its circumference travels is also <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=C+=+pi%2AD ALIGN=MIDDLE  ALT=\"C+=+pi%2AD\"   DISABLED_style= \"max-width:90%; height:auto;\" >?\"<BR>\n" );
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document.write( "If the front wheel makes two rotations its circumference distance is doubled and the<BR>\n" );
document.write( "distance it covers in those two rotations is:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%2AC%5Bf%5D+=+2+%2A+pi%2AD%2F2 ALIGN=MIDDLE  ALT=\"2%2AC%5Bf%5D+=+2+%2A+pi%2AD%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Note that in the right side the multiplier of 2 cancels the denominator of 2 so the equation<BR>\n" );
document.write( "becomes:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%2AC%5Bf%5D+=+pi%2AD ALIGN=MIDDLE  ALT=\"2%2AC%5Bf%5D+=+pi%2AD\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "and the right side of this equation is equal to the distance that the bicycle has traveled<BR>\n" );
document.write( "with one rotation of the rear wheel.  Therefore, for every rotation of the rear wheel the<BR>\n" );
document.write( "front wheel must rotate twice.  <BR>\n" );
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document.write( "Hope this explanation helps to clarify the problem a little bit for you.\r<BR>\n" );
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