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Topic: Snow Plough (Read 916 times) |
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ThudnBlunder
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Snow Plough
« on: Feb 17th, 2009, 2:23am » |
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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?
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towr
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Re: Snow Plough
« Reply #1 on: Feb 17th, 2009, 2:32am » |
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What's the relation between the depth of snow and the rate at which the plough can clear it??
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Grimbal
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Re: Snow Plough
« Reply #2 on: Feb 17th, 2009, 3:02am » |
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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?
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JiNbOtAk
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Re: Snow Plough
« Reply #3 on: Feb 17th, 2009, 5:19am » |
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11.00 am
A really wild guess, since I've never experienced snow before.
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ThudnBlunder
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Re: Snow Plough
« Reply #4 on: Feb 17th, 2009, 8:28am » |
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on Feb 17th, 2009, 2:32am, towr wrote:| What's the relation between the depth of snow and the rate at which the plough can clear it?? |
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Volume of snow removed per unit time = constant.
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ThudnBlunder
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Re: Snow Plough
« Reply #5 on: Feb 17th, 2009, 8:36am » |
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on Feb 17th, 2009, 3:02am, Grimbal wrote:| Can the plough actually clear an area while the snow is falling? |
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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?
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| « Last Edit: Feb 17th, 2009, 8:37am by ThudnBlunder » |
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towr
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Re: Snow Plough
« Reply #6 on: Feb 17th, 2009, 9:23am » |
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As far as I can tell, it started snowing 1/phi hours before noon
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
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| « Last Edit: Feb 17th, 2009, 9:25am by towr » |
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Immanuel_Bonfils
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Re: Snow Plough
« Reply #7 on: Feb 17th, 2009, 11:05am » |
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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)
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.
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towr
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Re: Snow Plough
« Reply #8 on: Feb 17th, 2009, 11:37am » |
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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.
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Immanuel_Bonfils
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Re: Snow Plough
« Reply #9 on: Feb 17th, 2009, 1:14pm » |
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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?
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| « Last Edit: Feb 17th, 2009, 1:17pm by Immanuel_Bonfils » |
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towr
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Re: Snow Plough
« Reply #10 on: Feb 17th, 2009, 2:11pm » |
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 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.
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Grimbal
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Re: Snow Plough
« Reply #11 on: Feb 18th, 2009, 5:05am » |
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on Feb 17th, 2009, 8:36am, 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?
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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.
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ThudnBlunder
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Re: Snow Plough
« Reply #12 on: Feb 18th, 2009, 6:55am » |
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on Feb 18th, 2009, 5:05am, Grimbal wrote:
In my opinion the road isn't clear if part of the road is covered with snow again.
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Semantically true. But this is a simple puzzle, not a Politburo Special Directive for the gulags.
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| « Last Edit: Feb 18th, 2009, 7:42am by ThudnBlunder » |
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Grimbal
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Re: Snow Plough
« Reply #13 on: Feb 18th, 2009, 7:56am » |
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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.
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tohuvabohu
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Re: Snow Plough
« Reply #14 on: Feb 18th, 2009, 10:34am » |
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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.
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towr
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Re: Snow Plough
« Reply #15 on: Feb 18th, 2009, 10:43am » |
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on Feb 18th, 2009, 10:34am, 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. |
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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.
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tohuvabohu
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Re: Snow Plough
« Reply #16 on: Feb 18th, 2009, 11:22am » |
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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.
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ThudnBlunder
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Re: Snow Plough
« Reply #17 on: Feb 19th, 2009, 6:07am » |
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on Feb 17th, 2009, 9:23am, towr wrote:
The time at which the plough started: t0 = sqrt(5)/2-1/2
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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.
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ooka
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Re: Snow Plough
« Reply #18 on: Feb 19th, 2009, 12:41pm » |
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hello
plz check my solution, thx
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
btw, thx for running this site
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| « Last Edit: Feb 19th, 2009, 2:32pm by ooka » |
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Eigenray
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Re: Snow Plough
« Reply #19 on: Feb 19th, 2009, 4:04pm » |
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on Feb 19th, 2009, 12:41pm, 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 |
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It looks like you are assuming the plow moves at a constant speed. But it slows down as the snow gets thicker.
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ooka
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Re: Snow Plough
« Reply #20 on: Feb 22nd, 2009, 2:47pm » |
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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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| « Last Edit: Feb 22nd, 2009, 2:49pm by ooka » |
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