Snow starts falling before noon on a cold December day.

analysis/snow Added material

Problem

Snow starts falling before noon on a cold December day.  At noon a
snow plow starts plowing a street.  It travels 1 mile in the first hour,
and 1/2 mile in the second hour.  What time did the snow start
falling??

You may assume that the plow's rate of travel is inversely proportional
to the height of the snow, and that the snow falls at a uniform rate.

Show solution

11:22:55.077 am.

Method:

Let b = the depth of the snow at noon, a = the rate of increase in the
depth.  Then the depth at time t (where noon is t=0) is at+b, the
snowfall started at t_0=-b/a, and the snowplow's rate of progress is 
ds/dt = k/(at+b).

If the snowplow starts at s=0 then s(t) = (k/a) log(1+at/b).  Note that
s(2 hours) = 1.5 s(1 hour), or  log(1+2A/b) = 1.5 log(1+A/b), where
A = (1 hour)*a.  Letting x = A/b we have (1+2x)^2 = (1+x)^3.  Solve for
x and t_0 = -(1 hour)/x.

The exact answer is 11:(90-30 Sqrt[5]).

_American Mathematics Monthly_, April 1937, page 245
E 275.  Proposed by J. A. Benner, Lafayette College, Easton. Pa.

The solution appears, appropriately, in the December 1937 issue,
pp. 666-667.  Also solved by William Douglas, C. E. Springer,
E. P. Starke, W. J.  Taylor, and the proposer.

See R.P. Agnew, "Differential Equations," 2nd edition, p. 39 ff.

Additional material

[ Editor: from mail to Chris Cole, May-June 1998, answering how R. P.
Agnew ("RPA") was known to have originated the problem. ]

I used the book as a reference when I taught a recitation section of
differential equations.  I recall reading an article in an MAA
publication, I think either The College Math Journal or Mathematics
Magazine (which are refereed publications) in which the author, maybe,
generalized the problem or something along those lines.  Anyway, I
definitely remember that the problem was attributed to RPA and the
reference was his text, because I already had known about the problem
and had used the text.

There are some important references and historical information in the
current The American Mathematical Monthly, Volume 105, Number 6,
June-July 1998 in an article by David Singmaster that reviews: After
Math, Puzzles and Brainteasers by Ed Barbeau, The Chicken from Minsk by
Yuri B. Chernyak, New Mathematical Diversions, revised edition, by
Martin Gardner.

-- Nathaniel Silver <mathelp@worldnet.att.net>