One morning it starts to snow at a constant rate. Later, at 6:00am, a snow plow sets out to clear a straight street. The plow can remove a fixed volume of snow per unit time.
If the plow covered twice as much distance in the first hour as the second hour, what time did it start snowing?
Defining some variables;
s=snow depth at 6:00 am
r=rate of snowfall (in per hours)
t=time in hours (primary variable)
v=linear velocity of plow
k=proportional constant (likely unneccesary)
Using common formula for distance -> d=v*t
Assume plow blade width is constant, therefore plow speed dependant only on snow height (this is questionable, as once the snow height rises to greater than the blade width the problem changes)
Creating equations...
Height of snow at discreet time h(t)=s+rt
v=k/h(t); v=k/(s+rt)
and therefore
d=∑[(kt)/(s+rt)], where the sum is an integral.
Now the distance travelled in hour 2 is half that done in hour one so we have ;
0.5*∑[(kt)/(s+rt)]{0 to 1}=∑[(kt)/(s+rt)]{1 to 2}
Lacking an integral table or the desire to solve this with my restricted memory, this is as far as I go. The answe should provide some relation between 's' and 'r', which can then be used to determine the time when the snow began...
Of course, calculus was never my strong point, so poke holes at will!
equation city
