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## Main Question or Discussion Point

**Trouble with "The Snowplow Problem"**

"One morning it began to snow very hard and continued snowing stadily throughout the day. A snowplow set out at 8:00AM to clear a road, clearing 2 miles by 11:00AM and an additional mile by 1:00 PM. At what time did it start snowing?"

This my my approach.

In the problem it states that we can assume the rate of snowfall is constant and that the speed at which the plow goes is inversely proportional to the height of the snow.

Using this information, the differential equation I come up with is..

[tex]\frac{dx}{dt} = \frac{k}{S}[/tex]

Where k = proportionality constant, and S = the depth of the snow.

Since we can assume the rate of snowfall is constant, it yields another differential equation...

[tex]\frac{dS}{dt} = C[/tex]

Solving for S..

[tex]\frac{dS}{dt} = C \rightarrow dS = Cdt \rightarrow S = Ct+D

[/tex]

Substituting this into the first equation and solving..

[tex]\frac{dx}{dt} = \frac{k}{Ct+D} \rightarrow x = \frac{k}{C}ln|Ct+D|+E[/tex]

My next guess would use (t=8 , x=0) (t=11 , x=2) and (t=13 , x=3) as initial values to solve for C, D and E. I get VERY messy results however. With the help of my calculator I get an numerical answer for t to be equal to -1.7085 or 11:18PM of the previous day. This doesn't make sense considering it states it starts to snow in the MORNING. if the time were positive it would translate to 1:42AM. Any help would be appreciated.

~Matt

EDIT: The chapter this is in is about solving first order differential equations.