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        riddles >> easy >> Snow Plough
        
(Message started by: ThudanBlunder on Feb 17th, 2009, 2:23am)
Title: Snow Plough
Post by ThudanBlunder on Feb 17th, 2009, 2:23am
Snow begins to fall in the morning and continues to fall at a constant rate all day. At midday a snow plough begins to clear a path for the city tram. It manages a kilometre during the first hour and half a kilometre during the second hour.

At what time did it begin to snow?
Title: Re: Snow Plough
Post by towr on Feb 17th, 2009, 2:32am
What's the relation between the depth of snow and the rate at which the plough can clear it??
Title: Re: Snow Plough
Post by Grimbal on Feb 17th, 2009, 3:02am
Can the plough actually clear an area while the snow is falling?  When it finishes one kilometer, wouldn't the other end already start to cover with snow?  Or is it one high-tech satellite-based microwave "plough" that clears an area uniformly from top to bottom until it is clear everywhere?
And when it proceeds with the 1/2 kilometer, does it still keeps the 1st kilometer clear?
Title: Re: Snow Plough
Post by JiNbOtAk on Feb 17th, 2009, 5:19am
[hide]11.00 am[/hide]

A really wild guess, since I've never experienced snow before.
Title: Re: Snow Plough
Post by ThudanBlunder on Feb 17th, 2009, 8:28am

on 02/17/09 at 02:32:41, towr wrote:
What's the relation between the depth of snow and the rate at which the plough can clear it??

Volume of snow removed per unit time = constant.
Title: Re: Snow Plough
Post by ThudanBlunder on Feb 17th, 2009, 8:36am

on 02/17/09 at 03:02:09, Grimbal wrote:
Can the plough actually clear an area while the snow is falling?  

As the plough wasn't manufactured in a Communist country, I don't see why not.

How is what happens to the cleared path relevant to the question?
Title: Re: Snow Plough
Post by towr on Feb 17th, 2009, 9:23am
As far as I can tell, it started snowing [hide]1/phi[/hide] hours before noon

[hide]
depth of snow: d(t) = r*t
speed of the plough: s(t) = c/d(t)
length of road the plough cleared: x(t)=int s(t) dt

x(t)=int c/r 1/t dt = c/r ln(t)

x(t0+1)-x(t0) = 2*(x(t0+2)-x(t0+1))
ln((t0+1)/t0) = 2 ln((t0+2)/(t0+1))
(t0+1)/t0 = (t0+2)2/(t0+1)2

The time at which the plough started: t0 = sqrt(5)/2-1/2
[/hide]
Title: Re: Snow Plough
Post by Immanuel_Bonfils on Feb 17th, 2009, 11:05am
As a matter of fact we have to overlook that , in each dx of the path,  it takes an theoretically infinite time to be cleared, since it's still snowing (brrrrr, like London? Politics apart I think more or less is what Grimbal had in mind)

[hide]towr made a slight confusion with time (realy a confusing matter); the time at which the plough started is given as noon. Let's call it t=0.
   Actually (using some of towr's symbols)  d(t)=r*T + r*t    where  T is the time interval of snowing before noon.
 So   x(t)= (c/r) ln((t+T)/T);
It should give t = towr answer rather as the time from the beggining of the snow till noon.
[/hide]
Title: Re: Snow Plough
Post by towr on Feb 17th, 2009, 11:37am
I don't know why you find it confusing. t=0 is when it starts to snow, and t0 is when the plough starts.
You're not making the computation any easier by adding extra constants.
Title: Re: Snow Plough
Post by Immanuel_Bonfils on Feb 17th, 2009, 1:14pm
Excuse me; my confusion. I didn't get that your t0 is referred to the beginning of the damned snow.

May be because the first  obs. : 1/phi hours?
Title: Re: Snow Plough
Post by towr on Feb 17th, 2009, 2:11pm
?!? Now I'm confused.

My timeline goes like this
t=0 it starts to snow
t0=1/phi, the plough starts to clear snow, this is noon
t1=t0+1=phi, the plough has cleared the first kilometer, 1 pm
t2=t0+2=phi2, the plough has cleared the next half kilometer, 2 pm

phi = sqrt(5)/2+1/2, is the golden ratio.
Title: Re: Snow Plough
Post by Grimbal on Feb 18th, 2009, 5:05am

on 02/17/09 at 08:36:09, ThudanBlunder wrote:
As the plough wasn't manufactured in a Communist country, I don't see why not.

How is what happens to the cleared path relevant to the question?

In my opinion the road isn't clear if part of the road is covered with snow again.
Actually, according to towr's calculations, at 1:00 part of the "cleared" road is covered with more snow than it was at noon.

But well, in a burocratic state, the road is clear of snow iff the plough went over the road, regardless of whether it is again covered with snow or whether there was any snow before the plough started.
Title: Re: Snow Plough
Post by ThudanBlunder on Feb 18th, 2009, 6:55am

on 02/18/09 at 05:05:00, Grimbal wrote:
In my opinion the road isn't clear if part of the road is covered with snow again.

Semantically true. But this is a simple puzzle, not a Politburo Special Directive for the gulags.
Title: Re: Snow Plough
Post by Grimbal on Feb 18th, 2009, 7:56am
I have a bias becaus it reminds me of the grazing cows problem.  There, the grass is consumed uniformly and the field is empty when there is no grass left.
Title: Re: Snow Plough
Post by tohuvabohu on Feb 18th, 2009, 10:34am
Too bad he didn't start plowing one second after it started snowing. He could have plowed the whole town in an instant, and taken the rest of the day off.
Title: Re: Snow Plough
Post by towr on Feb 18th, 2009, 10:43am

on 02/18/09 at 10:34:17, tohuvabohu wrote:
Too bad he didn't start plowing one second after it started snowing. He could have plowed the whole town in an instant, and taken the rest of the day off.
If he started one second after it started snowing, he could clear about 8.5 kilometers in the first hour. So it depends on the size of the town.
Title: Re: Snow Plough
Post by tohuvabohu on Feb 18th, 2009, 11:22am
We're talking about a snowplow that can only cover one kilometer of road in an hour, while the snow levels are still pretty low. Either this is one incredibly heavy, unexpected snowfall, or the amount of tram track he can be expected to plow has to be much less than 8 kilometers.
The math is all beyond me. I almost said, if he started plowing before it started snowing, but then people would have complained about him exceeding the speed of light.
Title: Re: Snow Plough
Post by ThudanBlunder on Feb 19th, 2009, 6:07am

on 02/17/09 at 09:23:39, towr wrote:
[hide]The time at which the plough started: t0 = sqrt(5)/2-1/2
[/hide]

Well done, towr.  :) Of course, the trap here is to assume that the average depth during the 2nd hour must be twice what it was during the 1st hour, giving an answer of 11.30am.
Title: Re: Snow Plough
Post by ooka on Feb 19th, 2009, 12:41pm
hello

plz check my solution, thx
[hide]

t - time passed till the plow started to work
d - height of snow cap per hour
k - distance covered by the plow

d*t is the height of cap at 12.00

during an hour of the plowing the height increased by 1*d

so during one hour the plow cleared:

(d*t+(1*d)/2)*k of snow

similarly, from 13.00 to 14.00 the plow cleared:

(d*(t+1)+(1*d)/2)*(1/2)*k

now we have equation that should give the answer (didn't solve yet)

(d*(t+1)+(1*d)/2)*(1/2)*k= (d*t+(1*d)/2)*k

[/hide]

btw, thx for running this site
Title: Re: Snow Plough
Post by Eigenray on Feb 19th, 2009, 4:04pm

on 02/19/09 at 12:41:33, ooka wrote:
during an hour of the plowing the height increased by 1*d

so during one hour the plow cleared:

(d*t+(1*d)/2)*k of snow

It looks like you are assuming the plow moves at a constant speed.  But it slows down as the snow gets thicker.
Title: Re: Snow Plough
Post by ooka on Feb 22nd, 2009, 2:47pm
thx eigenray

yeah, the speed and  the amount of snow that the plow can handle was supposed to be constant

in my case the plow covered some amount of snow per hour, and started covering new amount

will think twice before posting


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