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How Long Is A Piece Of String?

Posted by Karl Sharman on May 24, 2011 – 11:12 am

A nice easy one to relieve the midweek brain-strain…

A cylinder 84 cm high has a circumference of 16 cm. A string makes exactly 7 complete turns round the cylinder while its two ends touch the cylinder’s top and bottom. How long is the string?

I want to see the working out behind this too!

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15 Responds so far- Add one»

  1. 1. DP Said:

    140 cm.
    I’ll break it down into its parts:
    First, 1 time around ends up 12 cm higher than the start point (84/7=12). Second, it goes 16 cm around.
    These are the 2 sides of a right triangle if you were to “unroll” the cylinder. (12=height, 16=base)
    So sqrt(12^2 + 16^2) will give us a length of 20, which is the straight-line path the string will take.
    Now simply multiply that by 7 wraps, and you get 140 cm.

  2. 2. John24 Said:

    140 cm I agree with DP assuming the string takes the shortest path.

    I used a similar method as DP except I used the cylinder’s original 84 cm height and multiplied the 16cm circumference by 7 to get 112cm circumference. I unrolled the cylinder and had the string complete the hypotenuse of the triangle.

    84^2 + 112^2 = 140^2

  3. 3. Tatiani Said:

    140cm
    the same solution, I just “open” the cylinder to draw the string’s path.

  4. 4. Chris Said:

    Hi Karl. I nearly posted this one myself.

  5. 5. NewMan Said:

    You didn’t say how big the string was or the overlap of the touch on each end.

    Assume the string is 12 cm in diameter, then as you wrap the string around the cylinder, the length is 7*16 or 112, given the slop of where such a large string “touches”. If there is some argument here about touching, after you have measured this string out, cut it into 7 equal lengths, wrap each length with first touching last and side of next touching side of last and again it will still be 112.

    But I actually like what DP said.

  6. 6. TheWyvern Said:

    N = L / 2 ⁄7 r

    7 = L / 2*(22/7)*8
    L = 352 cm

  7. 7. SP Said:

    I also came up with John24’s method.

  8. 8. Chris Said:

    Me too. 140 cm.

  9. 9. Karl Sharman Said:

    TheWyvern – unlike you to get this wrong? DP gets in very quickly with the right answer, and as most have said – “unroll” the cylider for an easy view of the problem!

    NewMan – The ends of the string were touching – no mention of overlap.

  10. 10. science guy Said:

    it dos not

  11. 11. Kli Kogsy, Denmark Said:

    Definitely 140 cm.

  12. 12. AK Said:

    80 cm

  13. 13. apollo Said:

    140 is the wrong answer…ppl….

  14. 14. apollo Said:

    if the question is other way round that there is a cylinder of 15 cm and 4cm is the circumference of the cylinder and a string is required to wrap around it at the interval of 3 cm so that whole cylinder is complete then in that case what will be your approach to solve the problem ?

  15. 15. Chris Said:

    Hi Apollo. Why do you say 140 cm is the wrong answer? By unwrapping the cylinder we can see that there’s a right angled triangle of base 7*16 = 112 cm and height 84 cm. So the hypotenuse is 140 cm.

    If I’ve understood your question correctly, the string wraps around the cylinder 15/3 = 5 times. So the length of the string is sqrt((5*4)^2 + 15^2) = 25 cm

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