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## Homework Statement

Early one morning it starts to snow. At 7AM a snowplow sets off to clear the road. By 8AM, it has gone 2 miles. It takes an additional 2 hours for the plow to go another 2 miles. Let t = 0 when it begins to snow, let x denote the distance traveled by the plow at time t. Assuming the snowplow clears snow at a constant rate in cubic meters/hour:

a) Find the DE modeling the value of x.

b) When did it start snowing?

## Homework Equations

N/A, but related course topics are for solving DiffEq's by separation of variables

## The Attempt at a Solution

Modeling the plow as a rectangle with a certain length L, and assuming that the snow height H is across the length, and the plow speed is V, then "Assuming the snowplow clears snow at a constant rate in cubic meters/hour" can be expressed as

## LHv = C##

where c is some constant. H and V are both functions of time.

Simplify by combining constants (K= C/L) and dividing by H gives

##v= K/H##

re-expressing v as the time derivative of distance x yields

## \frac{dx}{dt} = \frac{K}{H} ##

separation of variables yields

## dx= \frac{K*dt}{H}##

I want to integrate as I usually would in a separation of variable problem, but I can't figure out what to do with H (can it be modeled as a function of x?), or how to incorporate the initial conditions. I'm also confused about setting t= 0 when it starts to snow, because that seems like it would make it into a piecewise function. I can't tell if I'm missing any assumptions.

To answer part B, it seems like I would have to pretend the plow was already moving by 7am (as in pretend it was continuously moving beforehand, and had some certain speed at 7am to match the initial conditions), then look at what time the speed approaches infinity from the right.