document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1183864>Question 1183864</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A clock has hour and minute hands 1 and 1.6 inches long respectively.  At what rate are the ends of the hands approaching each other when the time is 2 o'clock? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #814426 by </B><A HREF=/tutors/aboutme.mpl?userid=robertb><B>robertb(5830)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=robertb><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=robertb><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=814426\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28NOTE%29 ALIGN=MIDDLE  ALT=\"highlight%28NOTE%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >:  As a response to @ikleyn, the problem is basically one on RELATED RATES in Calculus, which entails relating the distance\r<BR>\n" );
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document.write( "between the tips of the minute and hour hands with the rate of change of angle between them.  So this is an EASY problem.\r<BR>\n" );
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document.write( "I don't really need to justify my solution, as the student has already confirmed/acknowledged that the answer is as what I have given.\r<BR>\n" );
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document.write( "So NO CONCEPTUAL ERRORS, just a straightforward application of related rates.\r<BR>\n" );
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document.write( "So as far as solutions are concerned, ikleyn's solution is GARBAGE, and needs to be thrown out to the TRASH BIN.  \r<BR>\n" );
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document.write( "And wash hands afterwards, to prevent spread of virus. :)\r<BR>\n" );
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document.write( "P.S.:  In ikleyn's solution it put the hour hand as being represented by the unit vector (<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cos%28pi%2F3%29 ALIGN=MIDDLE  ALT=\"cos%28pi%2F3%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sin%28pi%2F3%29 ALIGN=MIDDLE  ALT=\"sin%28pi%2F3%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >) and the minute hand as (<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1.6%2A%28pi%2F30%29 ALIGN=MIDDLE  ALT=\"1.6%2A%28pi%2F30%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0 ALIGN=MIDDLE  ALT=\"0\"   DISABLED_style= \"max-width:90%; height:auto;\" >). <BR>\n" );
document.write( "(She even mad a mistake on the minute hand representation.) VERY, VERY BIG MISTAKES.  Remember. at exactly 2 o'clock, the angle between them is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=pi%2F3 ALIGN=MIDDLE  ALT=\"pi%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" >, or sixty degrees, SO AT THAT INSTANT, the <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=theta ALIGN=MIDDLE  ALT=\"theta\"   DISABLED_style= \"max-width:90%; height:auto;\" > value for the minute hand CANNOT be <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=pi%2F3 ALIGN=MIDDLE  ALT=\"pi%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" > at the same time as the hour hand.  So her argument is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28FLAWED%29 ALIGN=MIDDLE  ALT=\"highlight%28FLAWED%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "Try to read and understand the problem TWICE, THRICE,... And formulate a correct solution in cycles of 8. :)\r<BR>\n" );
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document.write( "To ikleyn, just let me know if you need help in Calculus, and I will gladly help you.\r<BR>\n" );
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document.write( "Use the cosine law to relate the lengths of the minute and hour hands with the distance <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=d ALIGN=MIDDLE  ALT=\"d\"   DISABLED_style= \"max-width:90%; height:auto;\" > between their tips and the angle <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=theta ALIGN=MIDDLE  ALT=\"theta\"   DISABLED_style= \"max-width:90%; height:auto;\" >, the angle between them.\r<BR>\n" );
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document.write( "Then we have \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=d%5E2+=+1%5E2+%2B+%288%2F5%29%5E2+-+2%2A1%2A%288%2F5%29%2Acos%28theta%29+=+89%2F25-%2816%2F5%29%2Acos%28theta%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "===> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=d%2A%28dd%2Fdt%29+=+%288%2F5%29%2Asin%28theta%29%2A%28d%28theta%29%2Fdt%29 ALIGN=MIDDLE  ALT=\"d%2A%28dd%2Fdt%29+=+%288%2F5%29%2Asin%28theta%29%2A%28d%28theta%29%2Fdt%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "At 2 o'clock, the minute and hour hands are exactly <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=pi%2F3 ALIGN=MIDDLE  ALT=\"pi%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" > radians (or 60 degrees) apart.  <BR>\n" );
document.write( "At that instant, the distance between their tips is 7/5 inches.  (You can easily verify this!)\r<BR>\n" );
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document.write( "We need to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=d%28theta%29%2Fdt ALIGN=MIDDLE  ALT=\"d%28theta%29%2Fdt\"   DISABLED_style= \"max-width:90%; height:auto;\" >, which is easily obtained as \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=pi%2F360+-+pi%2F30+=+-%2811pi%29%2F360 ALIGN=MIDDLE  ALT=\"pi%2F360+-+pi%2F30+=+-%2811pi%29%2F360\"   DISABLED_style= \"max-width:90%; height:auto;\" >.  \r<BR>\n" );
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document.write( "I will not elaborate on this fact but rather will just leave the reason for your mental exercise, including what the NEGATIVE sign means \r<BR>\n" );
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document.write( "===> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=dd%2Fdt+=+%281%2Fd%29%2A%288%2F5%29%2Asin%28theta%29%2A%28d%28theta%29%2Fdt%29+=%285%2F7%29%2A+%288%2F5%29%2Asin%28pi%2F3%29%2A%28-%2811pi%29%2F360%29+=+-0.095 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >, to 3 d.p.\r<BR>\n" );
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document.write( "Therefore the tips of the minute and hour hands are are getting near each other at the rate of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%280.095%29 ALIGN=MIDDLE  ALT=\"highlight%280.095%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > inch per minute.\r<BR>\n" );
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document.write( " </SPAN>\n" );
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