document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=202242>Question 202242</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Pleaseee Help me with the following question, I will be grately thankful.\r<BR>\n" );
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document.write( "A bicycle has two sets of gears, one at the pedals and one at the rear wheel. A typical 10-speed bicycle has two gears on the chain wheel and five gears on the rear wheel. The speed at which a bicycle travels depends on three independent factors:<BR>\n" );
document.write( "- The first is the speed at which the cyclist pedals to turn the front gear (measured in rotations per minute).<BR>\n" );
document.write( "- The second is the gear ratio from the front gear to the rear wheel (a ratio between the number of teeth on the rear gear).<BR>\n" );
document.write( "- The third is the size of the rear wheel (measured as the diameter of the wheel).<BR>\n" );
document.write( "a)Develop a formula that predicts the velocity in km/h for a bicyclist.(2 marks)<BR>\n" );
document.write( "b)How fast will a cyclist travel (in km/h) who is pedalling at 50 rpm with a 42-toothed gear on the front, a 14-toothed gear on the back, and a tire with a diameter of 26 inches?(2 marks)<BR>\n" );
document.write( "c)In what ways can a cyclist pedal more slowly but maintain the same speed?(1 mark) </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #152505 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=152505\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> let x = number of gear teeth in front.<BR>\n" );
document.write( "let y = number of gear teeth in back.<BR>\n" );
document.write( "let d = diameter of the back wheel.<BR>\n" );
document.write( "-----<BR>\n" );
document.write( "let kmph = distance traveled in kilometers per hour.<BR>\n" );
document.write( "let r = radius of the back wheel in inches = d/2<BR>\n" );
document.write( "let c = circumference of the back wheel in inches.<BR>\n" );
document.write( "let k = inch to kilometer ratio.<BR>\n" );
document.write( "-----<BR>\n" );
document.write( "circumference of a circle = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%2Api%2Ar%29+ ALIGN=MIDDLE  ALT=\"2%2Api%2Ar%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" >.  this is also the same as <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=pi%2Ad ALIGN=MIDDLE  ALT=\"pi%2Ad\"   DISABLED_style= \"max-width:90%; height:auto;\" > since r = d/2.<BR>\n" );
document.write( "-----<BR>\n" );
document.write( "rpm = number of revolutions of the front crank per minute.<BR>\n" );
document.write( "rph = number of revolutions of the front crank per hour = 60 * rpm.<BR>\n" );
document.write( "-----<BR>\n" );
document.write( "your formula looks like it should be:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+kmph+=+rph+%2A+%28x+%2F+y%29+%2A+%282%2Api%2Ar%29+%2A+k+ ALIGN=MIDDLE  ALT=\"+kmph+=+rph+%2A+%28x+%2F+y%29+%2A+%282%2Api%2Ar%29+%2A+k+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "-----<BR>\n" );
document.write( "x = 42<BR>\n" );
document.write( "y = 14<BR>\n" );
document.write( "d = 26<BR>\n" );
document.write( "r = d/2 = 13<BR>\n" );
document.write( "c = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%2Api%2Ar%29+ ALIGN=MIDDLE  ALT=\"2%2Api%2Ar%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "k = .0254/1000<BR>\n" );
document.write( "rpm = 50<BR>\n" );
document.write( "rph = 50 * 60 = 3000<BR>\n" );
document.write( "-----<BR>\n" );
document.write( "your formula becomes the following, using these values:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=kmph+=+3000+%2A+%2842%2F14%29+%2A+2%2Api%2A13%29+%2A+%28.0254%2F1000%29+ ALIGN=MIDDLE  ALT=\"kmph+=+3000+%2A+%2842%2F14%29+%2A+2%2Api%2A13%29+%2A+%28.0254%2F1000%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "solving this equation, you get an answer of:<BR>\n" );
document.write( "18.6723701 which is the speed of the bicycle in kilometers per hour.<BR>\n" );
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document.write( "here's how it works.<BR>\n" );
document.write( "the front crank is roating at 50 revolutions per minute.<BR>\n" );
document.write( "times that by 60 and it becomes 3000 revolutions per hour.<BR>\n" );
document.write( "the number of teeth in the front crank is 42 and the number of teeth in the rear crank is 14.  1 revolution of the front crank causes the rear crank to revolve 3 times (42/14).<BR>\n" );
document.write( "this means that 3000 revolutions per hour of the front crank results in 9000 revolutions per hour of the rear crank per hour.<BR>\n" );
document.write( "each revolution of the rear crank cause the rear wheel to revolve one time.<BR>\n" );
document.write( "the circumference of the rear wheel is a function of the diameter.  since <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=c+=+2%2Api%2Ar+ ALIGN=MIDDLE  ALT=\"c+=+2%2Api%2Ar+\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%2Ar ALIGN=MIDDLE  ALT=\"2%2Ar\"   DISABLED_style= \"max-width:90%; height:auto;\" > = d, c = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=pi%2Ad ALIGN=MIDDLE  ALT=\"pi%2Ad\"   DISABLED_style= \"max-width:90%; height:auto;\" > expresses the same relationship because <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%2Ar ALIGN=MIDDLE  ALT=\"2%2Ar\"   DISABLED_style= \"max-width:90%; height:auto;\" > = d.  bottom line is 1 revolution of the rear wheel causes the bicycle to travel <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%2Api%2A13 ALIGN=MIDDLE  ALT=\"2%2Api%2A13\"   DISABLED_style= \"max-width:90%; height:auto;\" > inches, or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=pi%2A26 ALIGN=MIDDLE  ALT=\"pi%2A26\"   DISABLED_style= \"max-width:90%; height:auto;\" > inches which is the same thing and equals 81.68...... inches.<BR>\n" );
document.write( "this means that 9000 revolutions of the rear wheel per hour causes the bicycle to move 9000 * 81.68.... inches = 735132.68 ... inches.<BR>\n" );
document.write( "the ratio of inches to kilometers is .0254/1000.<BR>\n" );
document.write( "multiplying 735132.68 ... inches by .0254/1000 results in 18.67237 kmph (kilometers per hour).<BR>\n" );
document.write( "-----<BR>\n" );
document.write( "the answer looks reasonable.<BR>\n" );
document.write( "the miles per hour would be approximately .625 * 18.67237 = 11.67 mph which is a reasonable speed to assume for a bicycle.<BR>\n" );
document.write( "-----<BR>\n" );
document.write( "if the cyclist wants to pedal more slowly and achieve the same speed, he can do this by two basic ways.<BR>\n" );
document.write( "1.  increase the ratio of the front gear teeth to the back gear teeth.<BR>\n" );
document.write( "2.  increase the diameter of the rear wheel.<BR>\n" );
document.write( "either one of these will increase the distance traveled per revolution of the front crank.<BR>\n" );
document.write( " </SPAN>\n" );
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