Ponder This Challenge - December 1998 - Removing snow

Ponder This Challenge:

As you leave your house in the morning, you can feel the portent of snow in the air. The weather report on the radio confirms your suspicions.

Sure enough, the snow begins to fall before noon-time, and falls at a constant rate. The city sends out its first snow plow at noon which begins removing snow at a constant rate (in cubic feet per minute.) At 1 P.M. it has gone 2 miles. At 2 P.M. it has gone 3 miles, and is still not retracing its path.

At what time did it start to snow?
Good luck!
FYI: Although this problem may appear simple at first, the solution is actually fairly complex.

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Solution

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  December 1998 Solution:

  In order to find out what time it began snowing, you must engage in a bit of calculus.

  Let h denote the number of hours that it had been snowing before noon, so that the snow started at hour 12-h. Let s be the constant rate of snowfall (in inches per hour), and let r the constant rate of removal (in cubic feet per hour). At time 12+t, that is, t hours after noon, the depth of the snow is s times (t+h). The speed of the snowplow (in miles per hour) is inversely proportional to the depth of the snow, in order to maintain a constant rate of removal. So the snowplow is travelling at speed where the constant c converts (inches times miles times width of the snowplow blade) to (cubic feet). Fortunately we don't need to know any of the constants c, r or s.

  Between 12:00 and 1:00 the snowplow travels two miles, so that
  
  Between 1:00 and 2:00, with deeper snow, it travels only one mile:
  
  Integrating 1/(t+h) gives a logarithm, so that we find

  which yields
  
  Because it started snowing before noon, h is positive. h=0.618 hours translates to 37 minutes, so it started snowing at 11:23am.

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