Let's see who can solve this one :)

It starts snowing some time before 10:00 in the morning, and it continues to snow all day at a constant rate. A snowplow leaves the county garage and begins to plow the main highway at 10:00 am. At 11:00 am a second plow leaves from the same place to plow the same road. At noon, a third snow plow leaves from the same place to plow again the same road. Each plow is identical, and each moves at a rate inversely proportional to the depth of the snow. Some time in the afternoon, the 2nd plow catches up to the first one. At the same moment, the third plow also catches up to the other two.

What time did it start snowing?


(and this time without cheating and looking on the net please :p )

Hard question. I think this will involve solving some sort of differential equation.

Let s be the distance the plow's have travelled from their starting locations.

Summary:

At 10:00 the first plow started moving and the IV when s=0 is t = t0.

At 11:00 the second plow began moving and the IV when s=0 is t = t0 + 1hour.

At 12:00 the third plow leaves and the IV when s=0 is t = t0 + 2hours.

Now obviously t0 is the time since the snow began falling and we have to solve three equations.

We cannot solve this until we tie in the rule that the velocity of the vehicles is inversely proportional to the depth of snow. However the depth of snow is related to the time since it began falling! v = ds/dt = 1/t. This is simply a separable equation (first-order diff equ). So t = t0e^s.

The second plow isn't exactly the same because 1 hour has passed (instead of the unknown elapsed time like the first plow). So v = ds/dt = 1/(t - t0e^s). Unfortunately I have no idea how to solve that diff eq.

However I know the third plow has v = ds/dt = 1/(t - (1 + t0(1 - s))e^s.



There, I started the ball rolling, lets see if anyone can solve these three diff equations simultaneously. Hint, you should find a value for t which can be subtracted from 10:00am to find the time the snow started. (My maths class only started differential equations last week so maybe if this remains unsolved for a few more days I could do it!)

When the first snowplow will start, we don't know how much snow there is (more than one hour or less?). Thus, if there's less than one hour, it's initial speed will be v0. Its speed will decrease as the snow is still falling and we can call V1 its speed after 1 hour of snow.
In fact, v0>v1 if the snow just began, v0=v1 if the snowplow started after 1 hour of snow and v0<v1 if it started after more than 1 hour of snow.


The second snowplow will start at a speed of v1. Then, its speed will be higher than the first snowplow's one (more than 1 houre of snow on the road). The problem is that this second snowplot's speed will increase as soon as it will have reach the point where the first snowplow's speed was equal to v1 (before this point, it's speed was decreasing).

That's the same with the third snowplow!!! Sorry, but I don't have enough mathematical knowledges (or they are too old) to solve such a puzzle!

it started snowing 09:00 in the morning.?

Everneo : no that's not correct

I'm stuck figuring the formula for the distance travelled by the second plow.

How do you integrate 1/(u - ln(u) + c)?

Scratch that last - it's not relevant.

How do I solve this DE?
y = dy/dx + ae^x

Hang on... I've looked up DE techniques, and found an integrating factor. Back soon!

seems like it would have to start just as the first snow plow begins plowing.

I get three equations relating the time (t) to the position (p) for each of the three plows.
Snow started falling at t=0.
Plow 1 started at t = t₀
Plow 2 started at t = t₀ + 1
Plow 3 started at t = t₀ + 2

At any time t after the plows have started:
Plow 1 has travelled p₁ units
Plow 2 has travelled p₂ units
Plow 3 has travelled p₃ units

t is in hours.
p is in units chosen to set a constant factor relating the rate of snowfall to plow power to 1.

The equations:
t = t₀e^(p₁)
t = t₀ + 1 - p₂t₀e^(p₂)
t = t₀ + 2 - p₃²t₀e^(p₃)

Now I'm stuck trying to solve for t₀ when p₁=p₂=p₃

I'm not 100% confident of my work so far... how are you going with it, oxymoron? (For both the second two DEs, there is an integrating factor of e^-s)

Integrating Factor technique (http://www.sosmath.com/diffeq/first/intfactor/intfactor.html)

I cheated and found the answer to check my equations. I'm not quite right - I've missed an exponential factor in some of my terms (I think I ignored a constant or two somewhere that I shouldn't have).

Great, thanks for that Pete!!! I was just taught that at uni today! What a coincidence, it let me work out the problem. I got somewhere around 9:30am. I think. I'd like to check that with some other people though.