document.write( "Question 1098081: At exactly 12 o'clock noon the hour hand of a clock begins to move at six times its normal speed, and the minute hand begins to move backwards at five sixths its normal speed. When the two hands next coincide, what will be the correct time? \n" ); document.write( "

Algebra.Com's Answer #712584 by ikleyn(52788) $\"\"$ $\"About$

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\n" ); document.write( "At exactly 12 o'clock noon the hour hand of a clock begins to move at six times its normal speed, and the minute hand
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document.write( "As everybody knows, the hour hand makes one full rotation in 12 hours.\r\n" );
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document.write( "Hence, the regular angular speed of the hour hand is  of 360 degs, or 30 degs per hour, or  0.5 degs per minute.\r\n" );
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document.write( "Again, as everybody knows, the minute hand makes one full rotation in 1 hour = 60 minutes. \r\n" );
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document.write( "Hence, the regular angular speed of the minute hand is 360 degs per hour,  or 6 degs per minute.\r\n" );
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document.write( "The equation for the angular position of the \"crazy\" hour hand is\r\n" );
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document.write( "    H(t) = 90 - (6*0.5)*t                     (1)\r\n" );
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document.write( "where 90 = 90 degs is its initial position in degrees at noon (\"vertically up\");  (6*0.5) is its \"crazy\" angular velocity \r\n" );
document.write( "and \"t\" is time (! the CORRECT time !).\r\n" );
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document.write( "The equation for the angular position of the \"crazy\" minute hand is\r\n" );
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document.write( "    M(t) =        (2)\r\n" );
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document.write( "where 90 = 90 degs is its initial position in degrees at noon (\"vertically up\");  ((5/6)*6) is its \"crazy\" angular velocity and \r\n" );
document.write( "\"t\" is time in minutes (! the CORRECT time !). ( ! Notice that the sign \"+\" in formula (2) reflects anti-clockwise rotation). \r\n" );
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document.write( "The condition that the hour and the minute hands coincide is\r\n" );
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document.write( "    M(t) = H(t) + 360 degs,   or    = 90 - (6*0.5)*t + 360   (3)\r\n" );
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document.write( "Simplify equation (3) step by step.\r\n" );
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document.write( "    5*t = -3*t + 360  ====>  5t + 3t = 360  ====>  8t = 360  ====>  t =  = 45 minutes.\r\n" );
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\n" ); document.write( "\n" ); document.write( "Answer. 45 minutes.   When the two hands next coincide, the correct time is 45 minutes after noon.\r
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\n" ); document.write( "If you want to see other similar solved problems of this type, look into the lesson\r
\n" ); document.write( "\n" ); document.write( "    - Advanced clock problems\r
\n" ); document.write( "\n" ); document.write( "in this site.\r
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\n" ); document.write( "\n" ); document.write( "Also, you have this free of charge online textbook in ALGEBRA-I in this site\r
\n" ); document.write( "\n" ); document.write( "    ALGEBRA-I - YOUR ONLINE TEXTBOOK.\r
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\n" ); document.write( "\n" ); document.write( "The referred lesson is the part of this textbook under the section \"Word problems\", the topic \"Travel and Distance problems\".\r
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document.write( "    This topic is  a unique and a real treasure collection  of Travel and Distance problems, \r\n" );
document.write( "    from simple introductory and regular problems to advanced and very (the most) advanced.\r\n" );
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\n" ); document.write( "\n" ); document.write( "Save the link to this online textbook together with its description\r
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\n" ); document.write( "\n" ); document.write( "Free of charge online textbook in ALGEBRA-I\r
\n" ); document.write( "\n" ); document.write( "https://www.algebra.com/algebra/homework/quadratic/lessons/ALGEBRA-I-YOUR-ONLINE-TEXTBOOK.lesson\r
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\n" ); document.write( "\n" ); document.write( "to your archive and use it when it is needed.\r
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