Teepee puzzle

I know this one is very similar to the Cone Mountain puzzle, but that one was difficult to word. Hopefully this one will be more clear.

There is a teepee with radius 3 meters and height 5 meters. You wish to stake a rope to anywhere on the base of the teepee, wrap it entirely around the teepee once somehow, and tie the other end to the stake where you started.

1. What is the minimum length of rope needed?
2. Assuming the minimum length is used, what is the closest the rope gets to the tip of the teepee?


Photo credit: Wikipedia

1. Please don't just plop a URL to a solution elsewhere until a winner here has been declared.
   
2. All those who have won a beer previously are asked to not post answers or solutions for 24 after this posting. Past winners who must chime in early, may PM me.
   
3. Beer to the first satisfactory answer and solution, subject to rule 2.
   
4. Please put answers and solutions in spoiler tags.
   

I have been hoarding teepees.

Sorry, wrong thread

Five meters is NOT the slant height, it's the height from the center of the base to the tip.

I redid the numbers to not need to mention the slant height, fearing many people may not know what that is.

Either the answer is 6pi or you mean the rope wraps up the teepee, reaching the peak, then drops to the starting point.

In that case, wouldn't we need to know the thickness of the rope?

Quote: DJTeddyBear

Either the answer is 6pi or you mean the rope wraps up the teepee, reaching the peak, then drops to the starting point.

There is a better answer in between those two extremes.

Quote:

In that case, wouldn't we need to know the thickness of the rope?

Just assume it's string. We don't need to muddy the water with rope thickness.

Quote: Wizard

Just assuming it's string. We don't need to muddy the water with rope thickness.

Whoa....

I just did a quick Google search for ‘rope’. The first two links were images of rope for sale. It came in size from 1/4” to 1”.

A similar Google search for ‘string’ and also had links to purchase it, but only one of them had a size. 1/8“.

Obviously a 1/8” string is gonna take 8 times the length to get the same lateral height as a 1” rope.

So my answer remains that there’s insufficient information.

Quote: ssho88

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Minimum length of rope = 11.6498m and it is 0.26545m away from the tip. I think DJTeddyBear should be the winner !

By the way, I think the teepee is stable and can withstand strong wind.

Congratulations! This is the correct answer. I put the dimensions of the teepee incorrectly, but your answer is right for what I asked.

I hate to belabor this problem, but what would the answers be if the radius of the teepee were 5 meters and the slant height (distance from edge of base to tip) was 20 meters?

OH! Now that I see other replies, I realize what this question is really asking:

With a cone of this size, what's the smallest circumference an ellipse can have where one edge is at the base of the cone?

For the record, I believe that would be an ellipse where it's angle of inclination is at right angles to the opposite side from where it touches the base.

That in mind, I have no idea how to calculate such an ellipse, I'm just happy I figured out what is really being asked — and how to properly phrase it. 😁


And for the record, I REALLY thought that the rope was supposed to continuously wrap up the outside of the teepee, so that viewed from above it looks like a spiral. Viewed from the side, I thought it kinda looks like this:



Now I hope you realize why I thought that the size of the rope was important. 🤪

Quote: DJTeddyBear

With a cone of this size, what's the smallest circumference an ellipse can have where one edge is at the base of the cone?

Yes, the rope wraps around just ONCE. You are correct that it will form an ellipse.

Quote: Wizard

I hate to belabor this problem, but what would the answers be if the radius of the teepee were 5 meters and the slant height (distance from edge of base to tip) was 20 meters?

...time to go to the pub (while they're still open)!

Quote: charliepatrick

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I think this forms a right angle triangle as the radius is 20 and the arc length 10 Pi (2 Pi R) so the angle is Pi/2 = 90 degrees. Thus the answer would be 20 SQRT(2).

Thanks, I agree. I'll reply by PM to your second point.