A string length

[Genady]

A string is wound evenly around a circular rod exactly four times. What is the length of the string if the length of the rod is 12 cm, and its circumference is 4 cm?

[mistermack]

Spoiler

20 cm ? 

 

Edited May 23, 2023 by mistermack

[Genady]

5 minutes ago, mistermack said:

  Reveal hidden contents

20 cm ? 

 

Yes. +1

[TheVat]

Spoiler

The relation between a circle and a sine wave seems relevant here.  The string winding (just take one of the four) must be seen in a Euclidean plane by rotating the tube some.   The amplitude is 4/2π? And period is... ugh, I am rusty.  It is not three, because of rotation of tube.  (final calculation would be arc length for one cycle of sine wave, multiply by four.)

Nope, I need to rethink this.  Wait, easy way is use helix formula,  L = sq rt(height squared + circumference squared.

So...sq rt of 3 sq + 4 sq = 5.  So answer is 20?

 

Drat!  He beats me to it by one minute!  😀

[mistermack]

Spoiler

My thinking was that if you call the rod a piece of paper rolled up four times, then unroll it and you have a right-angled triange.

Then just use the square on the hypoteneus to work it out. 

 

[TheVat]

3 minutes ago, mistermack said:

  Hide contents

My thinking was that if you call the rod a piece of paper rolled up four times, then unroll it and you have a right-angled triange.

Then just use the square on the hypoteneus to work it out. 

Spoiler

Yes that's how helix formula is derived.  Well done.

 

Using this spoiler box system on a small tablet is going to drive me insane.

LOL

[Genady]

15 minutes ago, mistermack said:

  Hide contents

My thinking was that if you call the rod a piece of paper rolled up four times, then unroll it and you have a right-angled triange.

Then just use the square on the hypoteneus to work it out. 

 

That's how I thought about it, too. 

Spoiler

One turn, like in @TheVat's explanation, is a rod of length 3. Unroll it and get a classical right triangle, 3-4-5. Times 4.