document.write( "Question 1198138: Between 6 p.m. and 7 p.m. the hands of a clock make a ninety-degree angle on two occasions. If Jenny always leaves home to walk her dog when the first ninety-degree angle is formed, and arrives home when the second is formed, how much time, in hours, does Jenny spend walking her dog every week? \n" ); document.write( "

Algebra.Com's Answer #831669 by ikleyn(52787) $\"\"$ $\"About$

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\n" ); document.write( "Between 6 p.m. and 7 p.m. the hands of a clock make a ninety-degree angle on two occasions.
\n" ); document.write( "If Jenny always leaves home to walk her dog when the first ninety-degree angle is formed,
\n" ); document.write( "and arrives home when the second is formed, how much time, in hours,
\n" ); document.write( "does Jenny spend walking her dog every week?
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document.write( "We will measure angles starting from the position of the hour and the minute hands vertically up, at 12:00 (midday, the noon).\r\n" );
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document.write( "The minute hand makes one full rotation in one hour, so its angular velocity is 360 degrees per hour, or  \r\n" );
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document.write( "    360/60 = 6 degrees per minute.\r\n" );
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document.write( "The hour hand makes one full rotation in 12 hours, so its angular velocity is 360 degrees per 12 hours, or\r\n" );
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document.write( "    360/12 = 30 degrees per hour = 30/60 = 0.5 degrees per minute.\r\n" );
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document.write( "At 6:00 pm, the hour hand   is in position 6*30 = 180 degrees from vertical position clockwise.\r\n" );
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document.write( "            The minute hand is in vertical position (= 0 degrees) at that time.\r\n" );
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document.write( "\"t\" minutes after 6:00 pm, the hour hand is in position 180 + 0.5t  degrees from vertical up;\r\n" );
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document.write( "                           the minute hand is in position 6t  degrees.\r\n" );
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document.write( "After 6:00 pm, the hour hand and the minute hand make the right angle for the first time, when  180 + 0.5t = 6t + 90  degrees.\r\n" );
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document.write( "From this equation, we find  \r\n" );
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document.write( "     180 - 90 = 6t - 0.5t,  or  5.5t = 90,  t =  = 16.3636... minutes.\r\n" );
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document.write( "After 6:00 pm, the hour hand and the minute hand make the right angle for the second time, when  6t = 180 + 0.5*t + 90  degrees.\r\n" );
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document.write( "From this equation, we find  \r\n" );
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document.write( "     6t - 0.5t = 180 + 90,  or  5.5t = 270,  t =  = 49.0909... minutes.\r\n" );
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document.write( "ANSWER.  Ian was out with his dog 49.0909 - 16.3636 = 32.7272 minutes = 32 minutes and 44 seconds (approximately.\r\n" );
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\n" ); document.write( "\n" ); document.write( "On clock problems, see my lessons\r
\n" ); document.write( "\n" ); document.write( " - Clock problems \r
\n" ); document.write( "\n" ); document.write( " - Advanced clock problems\r
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