PROBLEM #12 SOLUTION
Let the depth of snow at time t to be t units. The speed of the plow at time t will be 1/t. Define
t=0 as the time it started snowing and t=x the time the plow started.
The distance covered in the first hour is the integral from x to x+1 of 1/t dt. The antiderivative of
1/t is ln(t) so the total distance covered in the first hour is ln((x+1)/x).
By the same reasoning the distance covered in the second hour in ln((x+2)/(x+1)).
Using the fact that it the plow traveled twice as far in the first hour as the second: ln((x+1)/x) =
ln((x+2)/(x+1))2
Exp both sides and you have (x+1)/x = ((x+2)/(x+1))2.
Solving for x you get x=(51/2-1)/2, which is the number of hours that elapsed between the time it
started snowing and the snow plow left.
Therefore, it started snowing at (51/2-1)/2 hours before 6:00am, or 5:22:55am.