document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=785545>Question 785545</A>: <I><SPAN STYLE=\"word-break: break-all;\"> At what time after 8:00 PM<BR>\n" );
document.write( "will the second hand bisect the<BR>\n" );
document.write( "angle formed by the minute<BR>\n" );
document.write( "hand and hour hand?<BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #477693 by </B><A HREF=/tutors/aboutme.mpl?userid=KMST><B>KMST(5328)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=KMST><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=KMST><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=477693\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> The hour hand sweeps through <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=360%5Eo ALIGN=MIDDLE  ALT=\"360%5Eo\"   DISABLED_style= \"max-width:90%; height:auto;\" > in 12 hours (<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=12%2A60minutes=720minutes ALIGN=MIDDLE  ALT=\"12%2A60minutes=720minutes\"   DISABLED_style= \"max-width:90%; height:auto;\" >)<BR>\n" );
document.write( "That angular velocity could be stated as<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=360%5Eo%2F%28%22720+minutes%22%29=0.5%5Eo ALIGN=MIDDLE  ALT=\"360%5Eo%2F%28%22720+minutes%22%29=0.5%5Eo\"   DISABLED_style= \"max-width:90%; height:auto;\" >per minute.<BR>\n" );
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document.write( "The minute hand sweeps through <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=360%5Eo ALIGN=MIDDLE  ALT=\"360%5Eo\"   DISABLED_style= \"max-width:90%; height:auto;\" > in 60 minutes<BR>\n" );
document.write( "That angular velocity could be stated as<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=360%5Eo%2F%28%2260+minutes%22%29=6%5Eo ALIGN=MIDDLE  ALT=\"360%5Eo%2F%28%2260+minutes%22%29=6%5Eo\"   DISABLED_style= \"max-width:90%; height:auto;\" >per minute.<BR>\n" );
document.write( " <BR>\n" );
document.write( "The second hand sweeps through <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=360%5Eo ALIGN=MIDDLE  ALT=\"360%5Eo\"   DISABLED_style= \"max-width:90%; height:auto;\" > in 1 minutes<BR>\n" );
document.write( "That angular velocity could be stated as<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=360%5Eo ALIGN=MIDDLE  ALT=\"360%5Eo\"   DISABLED_style= \"max-width:90%; height:auto;\" >per minute.<BR>\n" );
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document.write( "At 8:00:00 PM, the second hand amd the minute hand are pointing up, to 12, while the hour hand points to 8. They are forming 2 angles that add up to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=360%5Eo ALIGN=MIDDLE  ALT=\"360%5Eo\"   DISABLED_style= \"max-width:90%; height:auto;\" >:<BR>\n" );
document.write( "One is the <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=240%5Eo ALIGN=MIDDLE  ALT=\"240%5Eo\"   DISABLED_style= \"max-width:90%; height:auto;\" > angle containing the numbers from 1 through 7, and the other is the <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=120%5Eo ALIGN=MIDDLE  ALT=\"120%5Eo\"   DISABLED_style= \"max-width:90%; height:auto;\" > angle containing the numbers 9, 10, ans 11.<BR>\n" );
document.write( "After 8:00:00 PM, the second hand starts racing towards the hour hand, ahead of the slower minute hand.<BR>\n" );
document.write( "In 20 seconds it is 8:00:20 PM.<BR>\n" );
document.write( "The second hand has swept <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=120%5Eo ALIGN=MIDDLE  ALT=\"120%5Eo\"   DISABLED_style= \"max-width:90%; height:auto;\" > and is pointing at 4; the minute hand has swept only <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%5Eo ALIGN=MIDDLE  ALT=\"2%5Eo\"   DISABLED_style= \"max-width:90%; height:auto;\" >, and the hour hand has barely moved half a hair.<BR>\n" );
document.write( "At that point the second hand is almost bisecting the angle (about <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=238%5Eo ALIGN=MIDDLE  ALT=\"238%5Eo\"   DISABLED_style= \"max-width:90%; height:auto;\" >) formed by the minute<BR>\n" );
document.write( "hand and hour hand.<BR>\n" );
document.write( "If we do not need to be too accurate, we could say that at that time it is bisecting the angle formed by the minute<BR>\n" );
document.write( "hand and hour hand.<BR>\n" );
document.write( "If we need to be more precise, we can calculate angles a s a funtion of time.<BR>\n" );
document.write( "With <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t ALIGN=MIDDLE  ALT=\"t\"   DISABLED_style= \"max-width:90%; height:auto;\" >= minutes since 8:00:00 PM, the angles, counted clockwise betweeen the 12 o'clock position and each hand are:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=S%28t%29=t%2A360%5Eo ALIGN=MIDDLE  ALT=\"S%28t%29=t%2A360%5Eo\"   DISABLED_style= \"max-width:90%; height:auto;\" > for the second hand,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=M%28t%29=t%2A6%5Eo ALIGN=MIDDLE  ALT=\"M%28t%29=t%2A6%5Eo\"   DISABLED_style= \"max-width:90%; height:auto;\" > for the minute hand, and<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=H%28t%29=%288%2F12%29%2A360%5Eo%2Bt%2A0.5%5Eo=240%5Eo%2Bt%2A0.5%5Eo ALIGN=MIDDLE  ALT=\"H%28t%29=%288%2F12%29%2A360%5Eo%2Bt%2A0.5%5Eo=240%5Eo%2Bt%2A0.5%5Eo\"   DISABLED_style= \"max-width:90%; height:auto;\" > for the hour hand.<BR>\n" );
document.write( "To bisect bisecting the angle formed by the minute hand and hour hand, the second hand needs to form an angle of<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%281%2F2%29%28240%5Eo%2Bt%2A0.5%5Eo%2Bt%2A6%5Eo%29=%281%2F2%29%28240%5Eo%2Bt%2A6.5%5Eo%29=120%5Eo%2Bt%2A3.25%5Eo ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "(That is the average of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=M%28t%29 ALIGN=MIDDLE  ALT=\"M%28t%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=H%28t%29 ALIGN=MIDDLE  ALT=\"H%28t%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >).<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=S%28t%29 ALIGN=MIDDLE  ALT=\"S%28t%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > will be exactly that angle when<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t%2A360%5Eo=120%5Eo%2Bt%2A3.25%5Eo ALIGN=MIDDLE  ALT=\"t%2A360%5Eo=120%5Eo%2Bt%2A3.25%5Eo\"   DISABLED_style= \"max-width:90%; height:auto;\" >--><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t%2A360%5Eo-t%2A3.25%5Eo=120%5Eo ALIGN=MIDDLE  ALT=\"t%2A360%5Eo-t%2A3.25%5Eo=120%5Eo\"   DISABLED_style= \"max-width:90%; height:auto;\" >--><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t%2A356.75%5Eo=120%5Eo ALIGN=MIDDLE  ALT=\"t%2A356.75%5Eo=120%5Eo\"   DISABLED_style= \"max-width:90%; height:auto;\" >--><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t=120%2F356.75 ALIGN=MIDDLE  ALT=\"t=120%2F356.75\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "That is the time in minutes after 8:00:00PM.<BR>\n" );
document.write( "In seconds, it is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=60%2A120%2F356.75=21.1822 ALIGN=MIDDLE  ALT=\"60%2A120%2F356.75=21.1822\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "The second hand first bisects the angle formed by the minute hand and hour hand at 8:00:20.18 PM.<BR>\n" );
document.write( "It will happen again at about 8:00:50 PM, and again a little after 8:01:20 PM, and it will keep happening at intervals a hair longer than 30 seconds. </SPAN>\n" );
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