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Schoolkids and Snow

A group of schoolchildren is cleaning up the round rink from snow with shovels.

They managed to clean up the rink of 10 meters in radius in 1 hour, working in the optimal way.

How much time it would take them to clean up the rink of 20 meters in radius?

Dmitri Papichev
Monday, September 27, 2004

Isn't this just solving for the ratio of the areas?

Lou
Monday, September 27, 2004

I guess the answer is "no".

Dmitri Papichev
Monday, September 27, 2004

4 hours

PI * 10^2 in one hour,
PI * 20^2 in ??? hours.

msafar
Monday, September 27, 2004

The beauty of that problem is that most obvious answer (4 hours) is wrong.

Dmitri Papichev
Monday, September 27, 2004

If the way they clear the snow means that they just toss the snow to the side of the circle, as they widen the radius they have to re-shovel the snow that they just tossed to the side. Maybe the proper solution is to do take the ratio of the sum of the series.

So the sum of Pi*r^2 over r=0 to r=10 takes one hour.
If we find the sum or Pi*r^2 over r=0 to r=20 and divide it by
the sum from 0 to 10 that should might get the answer.  Does that sound right?

Unfortunately my math is so bad now that I don't remember how to solve for this. Plugging in numbers seems to give about 7.5 hours.

What kind of cruel taskmaster makes kids shovel snow for that long?

Howie
Monday, September 27, 2004

> If the way they clear the snow means that they just toss the snow to the side of the circle, as they widen the radius they have to re-shovel the snow that they just tossed to the side

That's exactly the case!

The math followed is indeed not so perfect :)

Dmitri Papichev
Tuesday, September 28, 2004

> The math followed is indeed not so perfect :)

Yep, my math is pretty weak :(  But it's been over 10 years since I've had to do anything more advanced than division...

Howie
Tuesday, September 28, 2004

The good tring is the problems doesn't require use of any advanced math :)

Dmitri Papichev
Tuesday, September 28, 2004

Hmmm, thanks for the hint. I think I've got it:

If you remove ever increasing radii of snow, the end volume of snow is like a half-sphere. Imagine stacking a disk of radius 1, on top of a disk of radius 2, on top of a disk of radius 3, ... If you have infinitesimally small disks, it basically works out to be a half-sphere.

So the volume of a half sphere is 2/3 * pi * r^3. 
For r = 10, we get 2000/3 * pi.
For r = 20, we get 16000/3 * pi.
Take the ratio and the radius of 20 should take 8 times as long as the radius of 10, so it should take 8 hours.

Technically, I guess it's probably not a perfect half-sphere, but the height radius of the half-sphere in this case seems like it's relative to the radius of the circle so the ratios would still come down to 10^3/20^3, I believe.

Howie
Tuesday, September 28, 2004

Yes, that's exactly correct. 8 hours.

The reasoning to get that answer could be as simple and clear as that:

The area of the larger rink is 4 times larger (proportional to R^2).
Each "elementary piece" of snow is to be moved away by the distance proportional to R (imagine two circles, small and large, and "map" every point of th small circle to the corresponding point of the large).
And the larger rink contains R^2 times more such "pieces".
Therefore, the total work to be done is proportional to R^3.

Of course, the same result could be obtained with the help of integrals, but we don't need to know anything about them.

Dmitri Papichev
Wednesday, September 29, 2004

I'd like to see some empirical data to support this? Volunteers?

B
Wednesday, September 29, 2004

Too many assumptions being made: the time taken is not necessarily proportional to the work done, and you can't give an accurate answer just with maths.

E.g. moving a 2kg weight from the center of a 10m circle to the radius does not take twice as long as moving a 1kg weight.

The time taken will depend among other things on the thickness of the snow, the technique being used, the size of the shovel, and the number and strength of the schoolchildren.

For example if you assume the snow is thin enough that one of the kids can shovel 40m of snow in one go, then the technique could be to shovel strips across the diameter of the circle.  The larger circle would take about twice as many such strips, each of which is about twice as long.  So approx 4 times as long.

But if the snow is 2m deep that's not going to work.

Joe
Friday, October 01, 2004

I disagree (Re: Too many assumptions being made).

All the reasonable assumptions apply here:
- the kids work "in optimal way", i.e. time is proportional to the work done, they are not getting tired, work with a constant efficiency, etc.;
- the rinks are big enough to not take into account geometrical parameters like shovel size and shape;
- the common technique of shoveling is used;
- the amount of snow is large enough to not have any edge cases like cleaning up al the area with a single shovel throw.

That makes the solution offered a quite reasonable one.

Also, that kind of tasks usually suggests some interactivity in finding a solution. If the readers asked for clarification, I would give them. But as long as there were no such requests, and the line of thought leading to the right answer was correct, I assumed the task is understood properly as it is :)

By the way, the perfect situation when 4 hours would be a correct answer is if the schoolkids used snow-melting machine, as there is no snow transfer in that case. But alas, they had only shovels :)

Dmitri Papichev
Monday, October 04, 2004

"Too many assumptions being made: the time taken is not necessarily proportional to the work done, and you can't give an accurate answer just with maths."

I like the question a lot. I don't care about the math or the correct answer - I'm interested that the applicant ask the questions about the process. Recognizing that the snow doesn't just evaporate, but has to be moved shows a thought process you want doing your business logic analysis (where you constantly ask "how" and "why")

Philo

Philo
Sunday, October 17, 2004

- the kids work "in optimal way", i.e. time is proportional to the work done, they are not getting tired, work with a constant efficiency, etc.;


Actually, if the kids are working so they arn't getting tired, them they arn't being optimal.  They can work harder to clear the rink faster as long as they don't collapse until the rink is cleared.  SO given that, if they really where working optimally to clear the 10' meter rink in an hour, they probably need to go WAY slower in order to have the energy left to do more clearing, so we would need to know the stanima of the kids...

Steamrolla
Monday, November 15, 2004

Ouch... sorry about that, my Word refuses to start, so that post was pretty bad.

Steamrolla
Monday, November 15, 2004

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