My solution to “How long is this coil of rope?”

See the original problem here

GReen coil of rope on white background
If the boards are nominally ‘2 by 6’s, how long is the visible portion of the rope?

My standard approach to modelling a problem like this is to load it into Geogebra. Then I look for various parts of the rope which resemble geometric entities I can calculate with.
It looks like a straight line will be an adequate model from F to E, and circles for the rest:

Straight enough to model with a line segment

I considered using Pythagoras theorem, but this is only worthwhile if you need practice with basic trigonometry.

Position points that you are giong to use for measurements as accurately as possible by zooming in, to minimise errors. Remember that it is the “dead-center’ of points where you will actually measure to.
Pro-tip: using crosses instead of circles to mark points is more precise. Next time I will!

3 points well spaced along the circumference in order to define 2 chords: FG and FH. They do not need to be visible as lines for us to be able to bisect them next:

Theorem: The center of a circle always lies along the bisector of any chord.

If you apply this theorem twice, you can find the exact center of any circle!
Here I used Geogebra’s Line-Bisector tool twice, to locate the center of the circle.

Why does this technique work? Point I is the point of intersection of the perpendicular bisectors of chords FG and FH. Point I has to be the center of the circle as we know it lies somewhere on both.

I used the “Circle with Center & Radius” tool and colored it orange, also making it thicker and easier to see. I did this to compare the shape of the coil to a circle centered on point I. Although not perfect, it looks circular enough to justify using circles to model it.

Can you see anywhere that the circle model is not a good fit?

Measuring the thickness of a single rope strand by carefully positioning points J & K.

IH = radius of the model circle in this image:

Next we divide the total radius of the circle by the thickness of one strand to find how many layers there are in the coil:

Radius / strand thickness = 2.08 / 0.25 = 8.32

We will round down and say there are about 8 circles – we could have just counted them of course but I like to use more general methods when possible, so that we could adapt this method for more difficult situations. Go and count now to double-check though…

Yes, there are 8.

We could model the situation as 8 separate circles, with radiuses beginning at 2.00 and reducing by 0.25 units each time. The total length of rope would approximately be equal to the sum of their circumferences.

In fact this is what I will do. It would of course be more accurate to model the coil as a spiral. In this recreational situation however it would not make enough difference to make me want to try it… If I had the actual length to compare my answer to, and I wanted to get as close as possible, this is what I would do.

Here are the radii and Circumferences of the 8 circles.

  Radius (r) Circumference (2 *pi *r)
1 2 4*pi
2 1.75 3.5*pi
3 1.5 3*pi
4 1.25 2.5*pi
5 1.00 2*pi
6 0.75 1.5*pi
7 0.5 1 * pi
8 0.25 0.5 * pi
Total   18 * pi

So the total length of the coiled section is about:

18 * Pi = 56.5 units

Plus the straight section EF which was 5.93 gives us:

56.5 + 5.9 = 62.4 units

But how long is a unit?

Deck boards are usually a little thicker than regular ‘1 by x’ lumber to make them feel firmer underfoot.

Well a single 1×4 board is 3.5 inches wide (I said you might want to look it up!), so let’s do this:

3.5 “/ 2.25 = 1.5556 inches per unit

So each unit is about 1-1/2 inches so we’ll multiply our answer by that conversion factor:

 62.4 * 1.5556 = 97.0694 inches

So let’s say about 97 inches.

With approximate calculations like this you should round your answers to make them easy to work with. Just be mindful of if you are rounding up or down and try to not drift too far!

To convert from a small unit to a larger one, always divide by the conversion factor. You will always end up with less of the BIG unit!

Next we divide inches by 12 to find how many feet:

97 / 12 = 8.0833 feet

My solution is: roughly 8 feet

What did you get? Write  a comment below.


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