document.write( "Question 332225: starting at 8am determine the number of times in the next 24 hours for which the hour and minute hands on a clock form a 180% angle \n" ); document.write( "
Algebra.Com's Answer #238269 by Edwin McCravy(20059)\"\" \"About 
You can put this solution on YOUR website!
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document.write( "The answer is 22 times in 24 hours.  Here's how to get it:\r\n" );
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document.write( "The hour hand makes 1 complete revolution in 12 hours.  That's\r\n" );
document.write( "\"360degrees%2F12hours\" or 30 degrees/hour.  The minute hand make 1 complete\r\n" );
document.write( "revolution in 1 hour, so that's 360 degrees/hour.\r\n" );
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document.write( "The angle between the hands is therefore increasing at the rate of\r\n" );
document.write( "(360-30) degrees/hour, or 330 degrees/hour.  Therefore in 24 hours\r\n" );
document.write( "the angle between the hands has increased by 24x330° or 7920°.\r\n" );
document.write( "Each revolution is 360° so we divide\"%227920%B0%22%2F%22360%B0%22\" and get 22 \r\n" );
document.write( "revolutions of the angle between the hands exactly.  Since the hands form \r\n" );
document.write( "a 180° angle exactly once each revolution, that's 22 times in 24 hours.\r\n" );
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document.write( "That's the answer, but for fun let's find those 22 times as a check.\r\n" );
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document.write( "At 8:00 the hands form a 120° angle.  Therefore the angle between the hands\r\n" );
document.write( "must increase by 60° in order to be 180° apart for the first time.  The angle\r\n" );
document.write( "between the hands increases at 330 degrees/hour, so it will take it 60/330 or\r\n" );
document.write( "2/11 ths of an hour for the hands to form a 180° angle.  2/11ths of an\r\n" );
document.write( "hour is \"%282%2F11%29%2A60\" or \"10%2610%2F11\" minutes after 8:00, or \r\n" );
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document.write( "8:10 10/11, the first time after 8:00 the hands will form a 180° angle.\r\n" );
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document.write( "\"%22360%B0%22%2F%22330%B0%22\" or \"12%2F11%7D%7D+or+%7B%7B%7B1%261%2F11\" hours later the hands\r\n" );
document.write( "will form a 180° angle again.  \"1%2F11\" of an hour is \"%281%2F11%29%2A60\" or\r\n" );
document.write( "\"5%265%2F11\" minutes, so that will be 1 hour and \"5%265%2F11\" minutes later which will be \r\n" );
document.write( "at 9:15 15/11 or 9:16 4/11.  \r\n" );
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document.write( "So here are the 22 exact times when the hands will form a 180° angle.\r\n" );
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document.write( " 1.  8:10 10/11\r\n" );
document.write( " 2.  9:16 4/11\r\n" );
document.write( " 3.  10:21 9/11\r\n" );
document.write( " 4.  11:27 3/11\r\n" );
document.write( " 5.  12:32 8/11\r\n" );
document.write( " 6.  1:38 2/11\r\n" );
document.write( " 7.  2:43 7/11\r\n" );
document.write( " 8.  3:49 1/11\r\n" );
document.write( " 9.  4:54 6/11\r\n" );
document.write( "10.  6:00\r\n" );
document.write( "11.  7:05 5/11\r\n" );
document.write( "12.  8:10 10/11\r\n" );
document.write( "13.  9:16 4/11\r\n" );
document.write( "14.  10:21 9/11\r\n" );
document.write( "15.  11:27 3/11\r\n" );
document.write( "16.  12:32 8/11\r\n" );
document.write( "17.  1:38 2/11\r\n" );
document.write( "18.  2:43 7/11\r\n" );
document.write( "19.  3:49 1/11\r\n" );
document.write( "20.  4:54 6/11\r\n" );
document.write( "21.  6:00\r\n" );
document.write( "22.  7:05 5/11\r\n" );
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document.write( "Edwin

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