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how to solve this word problem?

During my free time, I look for challenging math problems and try to solve them. But, the one I am going to show you is unsolvable.

(ex) A long strip of insulating tape is to be wrapped around a hot water pipe. If the width of the strip is w and the diameter of the pipe is d, at what angle must the strip be placed at the edge of the pipe so that no overlapping and no spaces occur?

I really cannot think of a way to do. Do you agree with me that it is unsolvable? If it is solvable, could someone explain how to do it? Thanks a lot.

phrontister

Real Member

From: The Land of Tomorrow

Registered: 2009-07-12

Posts: 4,880

Re: how to solve this word problem?

Hi Vietnamese, and welcome to the forum!

I found a trig solution...see image below, which hopefully is clear enough. If not, just let me know and I'll try to explain.

I've shown a 3.2-wide tape starting at bottom left of the front visible pipe face, moving upwards to the right (as indicated by the arrows) and wrapping around the vertical pipe 1.5 times. The broken lines show the location of the tape behind the visible pipe front.

The figures I've used are just examples, so enter your own into the formula to suit.

Formula to find angle GAH:

This next formula is shorter and will give exactly the same answer, but I think the proof might be a bit longer:

To ensure that the tape will cover the entire vertical pipe face, the top left corner of the tape must start at or below the bottom of the pipe.

Here is a video of an animation displaying the effect of different tape widths.

Last edited by phrontister (2017-05-23 16:12:45)

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"The good news about computers is that they do what you tell them to do. The bad news is that they do what you tell them to do." - Ted Nelson

davidtrinh

Member

Registered: 2016-12-03

Posts: 10

Re: how to solve this word problem?

Thank you very much for your detailed solution, phrontister. I really appreciate your time and I like your video.
You are a great help.

Last edited by davidtrinh (2017-05-23 19:47:24)

phrontister

Real Member

From: The Land of Tomorrow

Registered: 2009-07-12

Posts: 4,880

Re: how to solve this word problem?

Thanks, davidtrinh.

I did that in Geogebra, and had fun putting it together.

And I learnt some things along the way, too.

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"The good news about computers is that they do what you tell them to do. The bad news is that they do what you tell them to do." - Ted Nelson

Bob

Administrator

Registered: 2010-06-20

Posts: 10,623

Re: how to solve this word problem?

hi Phro,

I started in a similar way but decided to reject that method.  As you view the pipe sideways,  the diameter you 'see' is 6, but the tape is curving around the pipe, so a flat width of 3.2 is less when seen square on as you have shown ... I think.    There are formulas for working on a sphere so I'm sure there must be similar ones for a cylinder. I'm still at the research stage.

Bob

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Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

phrontister

Real Member

From: The Land of Tomorrow

Registered: 2009-07-12

Posts: 4,880

Re: how to solve this word problem?

Hi Bob,

I think I get your drift, but can't visualise it too well.

So...I got me a bit of round pipe and some duct tape, my vernier calipers and a compass, and did a prac test.

Pipe diameter = 29mm
Tape width = 18mm

I placed my compass at 72° to the side of the tube and drew a line, along which I then laid the top edge of the tape.

And...it worked!

Last edited by phrontister (2017-05-24 20:01:16)

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"The good news about computers is that they do what you tell them to do. The bad news is that they do what you tell them to do." - Ted Nelson

Bob

Administrator

Registered: 2010-06-20

Posts: 10,623

Re: how to solve this word problem?

hi Phro,

Experiment is good and hard to argue against.  I don't think my 'objection' amounts to a very big error anyway; it's the difference between the straight line connection between two points on a curved surface and the distance around the surface.  Unless the curvature is great it doesn't amount to much.  I tried to find a decent helix to make a diagram but no such luck.  Most of my googling ended up with DNA diagrams!

I'm going to try a 3-D equation of a helix and some coordinate geometry.  It may take some time as I'm up to my eyes in other things at the moment. 

Bob

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Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

phrontister

Real Member

From: The Land of Tomorrow

Registered: 2009-07-12

Posts: 4,880

Re: how to solve this word problem?

Hi Bob,

We only need to do 2-D straight-line cross-section measuring here, with any lines viewed from a point perpendicular to the front pipe face as if that face were flat.

If I cut through the pipe with my dropsaw, the result would be a straight-line cut, irrespective of whether I'd cut perpendicular to the pipe's sides (for the diameter), or cut parallel to the angled tape.

I think that straight-line measurements are sound for this problem, and that the presence of the curved surface is only an optical herring.

Of course, methods of calculating the length of tape needed to cover the pipe to a given height will differ between a flat surface and one that is curved.

Last edited by phrontister (2017-05-25 10:49:43)

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"The good news about computers is that they do what you tell them to do. The bad news is that they do what you tell them to do." - Ted Nelson

Bob

Administrator

Registered: 2010-06-20

Posts: 10,623

Re: how to solve this word problem?

hi Phro,

I agree that 2-D makes it easier. My first diagram was just like yours (except 90 round and mirror image).  My difficulty is with the width measurement of the tape.  To take an extreme case, if you stuck the tape along the length of the pipe and wrapped it around the sides as far as it would go, the W measurement is not the 2-D projection onto the diagram.  Say the tape went half way round; the W would be the semi-circumference; but it would 'look' like it was D wide (= W/pi).

Bob

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Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

phrontister

Real Member

From: The Land of Tomorrow

Registered: 2009-07-12

Posts: 4,880

Re: how to solve this word problem?

Hi Bob,

Yes, that is an extreme case, and one that is only a solution if the tape width is identical to the pipe circumference and the tape wraps widthwise around the pipe. It's probably not a valid illustration to use here, because all other solutions come from the tape wrapping lengthwise, in spiral fashion, around the pipe.

Another observation is that angle GAH cannot be less than 45°, which is shown on my video. The only exception to the minimum angle is that extreme case, where it is zero.

I still reckon the curved surface thing is a big, fat, red herring!

Btw, having seen some of your diagrams and explanations in geometry posts helped form the 2-D idea for me here.

You can see I'm fighting tooth and nail for my position: I just can't bear to think that my video may have missed the mark!

Last edited by phrontister (2017-05-25 03:23:54)

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"The good news about computers is that they do what you tell them to do. The bad news is that they do what you tell them to do." - Ted Nelson

phrontister

Real Member

From: The Land of Tomorrow

Registered: 2009-07-12

Posts: 4,880

Re: how to solve this word problem?

Hi Bob,

In my drawing in post #2, A and B are the outer points of the straight line AB that is both the diameter and the front curve of the vertical pipe. In 3-D, that dual straight-line property only occurs when AB is viewed along a plane that extends horizontally forward from AB...which is how the line is viewed in my 2-D model.

Similarly, AG, with G being directly above B, has that same dual straight-line property.

In such along-the-plane viewing, the 2-D model can be used, thus avoiding having to consider the curved nature of the pipe's surface.

That's how I see it.

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"The good news about computers is that they do what you tell them to do. The bad news is that they do what you tell them to do." - Ted Nelson

thickhead

Member

Registered: 2016-04-16

Posts: 1,086

Re: how to solve this word problem?

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{1}Vasudhaiva Kutumakam.{The whole Universe is a family.}
(2)Yatra naaryasthu poojyanthe Ramanthe tatra Devataha
{Gods rejoice at those places where ladies are respected.}

Bob

Administrator

Registered: 2010-06-20

Posts: 10,623

Re: how to solve this word problem?

hi thickhead,

Glad you're back on the case. 

Bob

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Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

phrontister

Real Member

From: The Land of Tomorrow

Registered: 2009-07-12

Posts: 4,880

Re: how to solve this word problem?

Hi all;

Sorry, thickhead, but I haven't looked at your method yet as I'm still struggling with my thoughts on this.

Anyway, I made another model that was much more accurate than before, and couldn't get it to agree with the straight-line method of my previous posts.

Using measurements from post #6 (pipe D=29, tape W=18 ), my model was giving an approx tape angle of 79°, so I had to rethink (sorry, Bob!)

I'd overlooked the fact that distance travelled governs the elevation angle, which is why my straight-line method (really a short cut) gave me the steeper angle (by about 7°).

I now have the following formula, for which I've chosen the upward angle from the side because it saves marking a line across if there's no pipe base to work from:

Edit: Hmmm...looking at thickhead's solution, I think I have a small error. Getting there, though...

Ah, yes...I inadvertently had GB as the tape width. This should be the correct formula:

Last edited by phrontister (2017-05-28 14:24:52)

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"The good news about computers is that they do what you tell them to do. The bad news is that they do what you tell them to do." - Ted Nelson