Hallway Pole
You are an architect. Your client, living in Flatland, wants a building designed with a long 3 foot wide corridor which opens into a larger hallway. You must design the hallway for the minimum width which will allow the inhabitants to move a 24 foot-long pole down the corridor and turn it into the hallway.
The corridor is perpendicular to the hallway. Since this is Flatland, the pole cannot be tilted up. The pole is rigid. How wide must the hallway be?
The corridor is perpendicular to the hallway. Since this is Flatland, the pole cannot be tilted up. The pole is rigid. How wide must the hallway be?
Labels: funphysics
posted by Rajesh Lal at
8:44 AM
18 Comments:
first
15.588 feet approx
Chris, if you are correct, which Im assuming you are, I would suggest you round it up to 16' to make measurements easier.
I always will quote to sufficient precision to remove any ambiguity.
Because I'm pretty good at maths and physics problems (I hate probability one's 'cos they can be very tedious), I'm trying to just give helpful clues when a puzzle is posted. In this case, rounding to 16 foot would be counter-productive because it is not the correct answer.
Besides, this is maths, I have no intention of actually getting my hands dirty measuring or building anything.
I could in fact have given an exact answer, but it might have put some people off.
We need to know the thickness of the walls and the width of the pole. I've never been to flatland, but if the walls and pole are near zero, then the hallway doesn't have to be wide at all.
Chris,
I don't know how you could arrive at 15+ feet without knowing the widths of the wall and the pole. I must be misunderstanding something. Did you assume these widths?
I figured if the walls and pole are both 0.25 foot in width, the hall only needs to be 3.75 feet wide. How thick does a wall with zero height need to be?
Ragknot - the problem only concerns with the interior width between the walls. The thickness of the walls is irrelevant.
I'm not sure how you're visualising the problem. I just see two corridors of different widths, butting at a rightangle - an "L" shape. The Flatland filler is just mind-candy for "this is a problem in plane geometry".
I have assumed the pole to have negligible thickness. As always with many of these types of problems, you get to the heart of the problem, and use common-sense to dispose of the trivia.
If the thickness of the pole did have to be allowed for, the problem would be significantly harder, I knew that straight away. I would have skipped the problem without trying.
I got 15.558 also
At first I thought the pole was going into a room with a 3 foot doorway, not two hallways.
I don't know where this 15.558 comes from.
It all depends how far the pole has to be "turned into" the hallway.
You could get it into a 3 ft wide hallway at a very narrow angle.
But if it has to be turned 90 degrees into the hallway then the hallway would have to be 24 ft wide.
So what am I missing?
Ragknot, now I understand why you were perplexed. Lesson: never add complications to the problem, unless you think that will provide a (preferably nice) technique for solving the original problem. I might post something to illustrate what I mean.
Last anonymous, you've got to get the pole completely into the hallway (which I think of as a wide corridor). You are taking the pole completely around a corner.
Chris, I repeat: the hallway needs to be no wider than the corridor if the pole only needs to be at a few degrees, but needs to be 24 ft wide if the pole has to swing round to 90 degrees in the hallway from the corridor. The depth of the hallway (beyond the end of the corridor) is not specified.
15.588 feet would allow you to have the pole at around 45 degrees from the corridor, but only provided the hall was of a similar depth.
Your last para is stating the obvious, but does not lead to 15.588.
15.558 feet let's you get the whole pole into the hall (but at an angle). I assumed the hallway to be infinitely long. It needs to be at least 18.27 feet to fully admit the pole and 24 feet to turn the pole through the complete 90 deg.
I know the week's not up.
It's also obvious that some of you can't see the geometry.
The following link should make it all very clear:
http://archives.math.utk.edu/visual.calculus/3/applications.2/
This post has been removed by the author.
This post has been removed by the author.
Although I assumed an "L" shape, the same results apply with a "T" shape.
Ah, now all is clear.
Another case of bad wording confusing the problem - unfortunately by no means the first time this has happened.
Had the problem referred to the DEPTH of the hallway rather than the WIDTH it would have been clearer.
Or, indeed put it in terms of a pole or ladder going round a right angled turn in a corridor.
Reminds me of a similar problem: what is the largest area of a sofa that can be carried round a corner of unit width? Can't remember the answer!
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