document.write( "Question 1092284: The time is past 2 o'clock. In 10 minutes, the minute hand will be as much ahead of the hour hand as it is now behind it. What time is it? \n" ); document.write( "

Algebra.Com's Answer #706893 by ikleyn(52814) $\"\"$ $\"About$

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document.write( "The way to solve this problem is to write equation for angles that the minute hand and the hour hand make with . . . let say, \r\n" );
document.write( "with the vertical direction, with the direction \"vertically up\".\r\n" );
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document.write( "So, we will count angles clockwise starting from that direction \"vertically up\".\r\n" );
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document.write( "The minute hand makes one full revolution in 1 hour = 60 minutes. Hence, its angular velocity is   = 6 degrees per minute.\r\n" );
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document.write( "The hour hand makes one full revolution in 12 hours. Hence, its angular velocity is 12 times less, i.e.  =  degrees per minute.\r\n" );
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document.write( "Let \"t\" be the number of minutes after 2:00 pm which is under the question.\r\n" );
document.write( "Now the minute hand is in position 6t degrees, while the hour hand is in position   degrees.\r\n" );
document.write( "Thus the angle between the hands is   degrees.  \r\n" );
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document.write( "      Notice that 60 = 60 degs corresponds to the position of the hour hand at 2:00 pm.\r\n" );
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document.write( "In 10 minutes, the minute hand will be in position of  6(t+10) degrees, while the hour hand will be in position of  degrees; \r\n" );
document.write( "thus the angle between the hands will be .\r\n" );
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document.write( "We should determine the time moment \"t\" by equating these two angles:\r\n" );
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document.write( " = .   (1)\r\n" );
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document.write( "To solve the equation (1), multiply both sides by 12. You will get\r\n" );
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document.write( "12*6*(t+10) - 60*12 - 6*(t+10) = 60*12 + 6t - 12*6t.\r\n" );
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document.write( "Simplify:\r\n" );
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document.write( "12*6*t + 12*6*10 - 60*12 - 6t - 60 = 60*12 + 6t - 12*6t,\r\n" );
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document.write( "12*6*t - 6t - 6t + 12*6*t = 60*12 - 12*6*10 + 60*12 + 60,\r\n" );
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document.write( "132t = 780  ====>  t =  = 5.909 minutes = 5 minutes 54.(54) seconds = 5 minute and 54.546 seconds (approximately). \r\n" );
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\n" ); document.write( "\n" ); document.write( "Answer. The time under the question is 2 hours 5 minutes and 54.546 seconds (approximately; correct up-to 3 decimals after the decimal dot).\r
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