Question 1176132
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One day snow begins to fall at a constant rate.
At noon, a snowplow starts clearing a road. The plow clears a constant volume per hour.
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At 1PM, it has cleared 2 miles.
At 2PM, it has cleared 1 more mile.
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What time did it begin to snow?
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            As usual,  the major point in such problem is to make a correct setup.


            For it,  you need a right idea and right mathematical model.


            And,  as usual,  only a person,  who is equally good in  Physics and  Mathematics  (like me)

            can do it in a right way.   (You may consider it as a joke,  although I am absolutely serious)



<pre>
Let start making a sketch.  In the Figure, I show the first 2 miles of the road (segment AB)
and the next 1 mile (segment BC)



   ---|-----------------------|-------------|------------

      A      (2 miles)        B  (1 mile)   C



Let "t" denotes the time (the duration in hours) of snowing BEFORE the noon.

Let U be the rate of snowing (= the volume of the snow per mile of the road per hour).

Let V be the rate of clearing the road (= the volume of the snow taken off by the machine per hour).


Values U and V are constant in time.


During the time of t hours, the part of the road AB obtained the volume of the snow equal to

    2 miles * U * t hours = 2Ut units of the snow volume.

During one hour from the noon to 1 pm, snowing added 

    2 miles * U * 1 hour  = 2U   units of the snow volume.



It was taken off by the machine in 1 hour, so we have this equation

    2Ut + 2U = V         (1)

It is our first equation.



During next hour from 1 pm to 2 pm, snowing added  the amount of snow  2U on the part AB of the road.

On the part BC of the road, we have at 2 pm the amount of snow equal to  Ut + 2U.

This amount of snow on AC, totaled   2U + (Ut + 2U), was taken off by the machine in 1 hour.

It gives us the second equation

    2U + (Ut + 2U) = V    (2)


Compare equations (1) and (2).  Their right sides are identical  --- hence, their left sides are equal

    2Ut + 2U = 2U + (Ut + 2U).


Simplify 

    2Ut - Ut = 2U + 2U - 2U

       Ut    = 2U


Cancel U in both side

        t     = 2.


<U>ANSWER</U>.  It started snowing 2 hours before noon, i.e. at 10 am.
</pre>

Solved.