document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1165986>Question 1165986</A>: <I><SPAN STYLE=\"word-break: break-all;\"> its nine o'clock . how long will it be before both hands are pointing in the same direction? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #790463 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=790463\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> a full cycle of the clock is 360 degrees.\r<BR>\n" );
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document.write( "the hour hand does this in 12 hours.<BR>\n" );
document.write( "therefore, the speed of the hour hand is 360 / 12 = 30 degrees per hour.\r<BR>\n" );
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document.write( "the minute hand does this in 1 hour.<BR>\n" );
document.write( "therefore, the speed of the minute hand is 360 degrees per hour.\r<BR>\n" );
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document.write( "rate * time = distance.\r<BR>\n" );
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document.write( "the rate is in degrees per hour.<BR>\n" );
document.write( "the time is in hours.<BR>\n" );
document.write( "the distance is in degrees.\r<BR>\n" );
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document.write( "when they start, the hour hand is at 9 * 30 = 270 degrees and the minute hand is at 0 degrees.\r<BR>\n" );
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document.write( "for the hour hand, rate * time = distance becomes 30 * T + 270 = D.<BR>\n" );
document.write( "for the minute hand, rate * time = distance becomes 360 * T = D.\r<BR>\n" );
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document.write( "note that 30 * T + 270 = D is equivalent to 30 * T = D - 270.<BR>\n" );
document.write( "this means the hour hand needs to move 270 degrees less than the minute hand for them to be equal to the same number of degrees on the clock.\r<BR>\n" );
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document.write( "the two equations that need to be solved simultaneously are:\r<BR>\n" );
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document.write( "30 * T + 270 = D<BR>\n" );
document.write( "360 * T = D\r<BR>\n" );
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document.write( "since they are both equal to D, then 30 * T + 270 = 360 * T<BR>\n" );
document.write( "subtract 30 * T from both sides of this equation to get:<BR>\n" );
document.write( "270 = 330 * T<BR>\n" );
document.write( "solve for T to get:<BR>\n" );
document.write( "T = 270 / 330 = .8181818181..... hours.\r<BR>\n" );
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document.write( "in .8181818181..... hours, .....\r<BR>\n" );
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document.write( "the hour hand will have moved from 270 degrees to 294.54545454..... degrees.<BR>\n" );
document.write( "the minute hand will have moved from 0 degrees to 294.54545454..... degrees.\r<BR>\n" );
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document.write( "they will both be pointing in the same direction on the clock.\r<BR>\n" );
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document.write( "the time will be 9 + .8181818181... = 9.8181818181 hours.<BR>\n" );
document.write( "that's equivalent to 9 hours and 49.0909090909... minutes.\r<BR>\n" );
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document.write( "round to the nearest minute and that becomes 9:49 on the clock.\r<BR>\n" );
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document.write( "round to the nearest second and that becomes 9:49:05 on the clock.\r<BR>\n" );
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document.write( "here's a visual display of what it looks like at 9:49:05 on the clock.\r<BR>\n" );
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document.write( "the hour hand is in orange.<BR>\n" );
document.write( "the minute hand is in green.<BR>\n" );
document.write( "you can't see the  hour hand because the minute hand is covering it.<BR>\n" );
document.write( "the pointer handle for the minute hand is further away from the center of the clock than the pointer handle for the hour hand.<BR>\n" );
document.write( "the pointer handles are the blue circles that you see on the hands of the clock.\r<BR>\n" );
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document.write( " </SPAN>\n" );
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