Let be time in hours since it started to snow.
Let the rate of snowfall be metres/hr
Let the plough swept width be metres.
let the distance of the plough from its starting point be km.
Then as the plough clears a fixed volume of snow per unit time we have:
where is the constant volume swept per hour in units of thousanths of a cubic metre/hr (but the units are unimportant). (this is the swept width times the snow depth times the rate of progress of the plough)
This is an ordinary differential equation of variables seperable type and has general solution:
for some constant , and times after which is how long Noon is after [tex]t=0[tex].
So from the given conditions we have:
The first of these equations gives:
Dividing the secon by the third equation elliminates , and can be rearranged to give:
Simplifying this last equation gives us: , which has solutions and . The first of these is negative and so a spurious solution, so:
So the snow began falling minutes before noon or about .