document.write( "Question 1176132: One day snow begins to fall at a constant rate.
\n" ); document.write( "At noon, a snowplow starts clearing a road. The plow clears a constant volume per hour.
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\n" ); document.write( "At 1PM, it has cleared 2 miles.
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\n" ); document.write( "What time did it begin to snow? \n" ); document.write( "

Algebra.Com's Answer #802221 by ikleyn(52814) $\"\"$ $\"About$

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\n" ); document.write( "One day snow begins to fall at a constant rate.
\n" ); document.write( "At noon, a snowplow starts clearing a road. The plow clears a constant volume per hour.
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\n" ); document.write( "At 1PM, it has cleared 2 miles.
\n" ); document.write( "At 2PM, it has cleared 1 more mile.
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\n" ); document.write( "What time did it begin to snow?
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\n" ); document.write( "\n" ); document.write( "            As usual, the major point in such problem is to make a correct setup.\r
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\n" ); document.write( "\n" ); document.write( "            For it, you need a right idea and right mathematical model.\r
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\n" ); document.write( "\n" ); document.write( "            And, as usual, only a person, who is equally good in Physics and Mathematics (like me)\r
\n" ); document.write( "\n" ); document.write( "            can do it in a right way. (You may consider it as a joke, although I am absolutely serious)\r
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document.write( "Let start making a sketch.  In the Figure, I show the first 2 miles of the road (segment AB)\r\n" );
document.write( "and the next 1 mile (segment BC)\r\n" );
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document.write( "      A      (2 miles)        B  (1 mile)   C\r\n" );
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document.write( "Let \"t\" denotes the time (the duration in hours) of snowing BEFORE the noon.\r\n" );
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document.write( "Let U be the rate of snowing (= the volume of the snow per mile of the road per hour).\r\n" );
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document.write( "Let V be the rate of clearing the road (= the volume of the snow taken off by the machine per hour).\r\n" );
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document.write( "Values U and V are constant in time.\r\n" );
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document.write( "During the time of t hours, the part of the road AB obtained the volume of the snow equal to\r\n" );
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document.write( "    2 miles * U * t hours = 2Ut units of the snow volume.\r\n" );
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document.write( "During one hour from the noon to 1 pm, snowing added \r\n" );
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document.write( "    2 miles * U * 1 hour  = 2U   units of the snow volume.\r\n" );
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document.write( "It was taken off by the machine in 1 hour, so we have this equation\r\n" );
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document.write( "    2Ut + 2U = V         (1)\r\n" );
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document.write( "It is our first equation.\r\n" );
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document.write( "During next hour from 1 pm to 2 pm, snowing added  the amount of snow  2U on the part AB of the road.\r\n" );
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document.write( "On the part BC of the road, we have at 2 pm the amount of snow equal to  Ut + 2U.\r\n" );
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document.write( "This amount of snow on AC, totaled   2U + (Ut + 2U), was taken off by the machine in 1 hour.\r\n" );
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document.write( "It gives us the second equation\r\n" );
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document.write( "    2U + (Ut + 2U) = V    (2)\r\n" );
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document.write( "Compare equations (1) and (2).  Their right sides are identical  --- hence, their left sides are equal\r\n" );
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document.write( "    2Ut + 2U = 2U + (Ut + 2U).\r\n" );
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document.write( "Simplify \r\n" );
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document.write( "    2Ut - Ut = 2U + 2U - 2U\r\n" );
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document.write( "       Ut    = 2U\r\n" );
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document.write( "Cancel U in both side\r\n" );
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document.write( "        t     = 2.\r\n" );
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document.write( "ANSWER.  It started snowing 2 hours before noon, i.e. at 10 am.\r\n" );
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