document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1165490>Question 1165490</A>: <I><SPAN STYLE=\"word-break: break-all;\"> After a woman riding a bicycle removes her feet from the pedals, the front wheel rotates 80 times during the first minute. Then In succeeding 1 minute period, the wheel rotates 3/5 as many as times as it did the previous period. Determine the number of rotations of thee wheel before it stops.\r<BR>\n" );
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document.write( "I got 32 by using the Infinite Geometric Sequence but I'm not too sure about my answer  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #789978 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=789978\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> front wheel rotates 80 times during the first minute.<BR>\n" );
document.write( "it rotates 3/5 as many times as it did the previous minute for the remaining minutes.<BR>\n" );
document.write( "i believe this is the sum of an infinite geometric series.\r<BR>\n" );
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document.write( "that formula is S = a / (1 - r)<BR>\n" );
document.write( "in that formula, S equals the number of rotations and a = 80 and 1 - r = 1 - 3/5 = 2/5 = .4<BR>\n" );
document.write( "the formula becomes S = 80 / .4 = 200.\r<BR>\n" );
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document.write( "this says the number of rotations will be 200 before the bicycle stops.\r<BR>\n" );
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document.write( "i used excel to determine how minutes it would take to reach 200 rotations.<BR>\n" );
document.write( "the results of that analysis is shown below:\r<BR>\n" );
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document.write( "the answer has to be greater than 80 since it started at 80 and there would only be additional rotations after that.<BR>\n" );
document.write( "the infinite geometric sequence formula is S = a / (1-r)<BR>\n" );
document.write( "in your problem, a = 80 and r = 3/5<BR>\n" );
document.write( "S = a/(1-r) becomes S = 80 / (2/5) = 80 * 5/2 = 40 * 5 = 200.<BR>\n" );
document.write( "the bike wheel will have rotated 200 times before stopping.\r<BR>\n" );
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document.write( "the finite geometric series formula is Sn = a * (1 - r^n) / (1-r)<BR>\n" );
document.write( "when a = 80, this becomes Sn = 80 * (1 - r^200) / (1-r)\r<BR>\n" );
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document.write( "you can see that, after 44 minutes have elapsed the number of rotations remains at 200, meaning that, effectively, the bike wheel has stopped rotating.\r<BR>\n" );
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document.write( "you can use the sum of a finite geometric series to see that this is true.<BR>\n" );
document.write( "that formula is Sn = a * (1 - r^n) / (1 - r)\r<BR>\n" );
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document.write( "when n = 44, formula becomes S44 = 80 * (1 - .6^44) / .4.<BR>\n" );
document.write( "solve for S44 to get:<BR>\n" );
document.write( "S44 = 200\r<BR>\n" );
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document.write( "it doesn't matter how many minutes are elapsed, the number of rotations will remain at 200.\r<BR>\n" );
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document.write( "assume 500 minutes have elapsed.<BR>\n" );
document.write( "the formula becomes S500 = 80 * (1 - .6^500) / .4<BR>\n" );
document.write( "solve for S500 to get:<BR>\n" );
document.write( "S500 = 200.\r<BR>\n" );
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document.write( "the number of rotations stops at 200 which means the bicycle has stopped rolling when 44 minutes have elapsed.\r<BR>\n" );
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document.write( "note that:<BR>\n" );
document.write( "r = 3/5 is the same as r = .6<BR>\n" );
document.write( "(1 - r) = (1 - 3/5) = 2/5 which is the same as 1-r = .4<BR>\n" );
document.write( "(1 - r^n) = (1 - (3/5)^n) = 1 - .6^n\r<BR>\n" );
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