# Thread: How long is the rope?

1. ## How long is the rope?

You have a rope that will reach around the earth (+/- 40,000 km). If you put sticks of 1 meter high around the world and lay the rope on these sticks, how much longer will the rope need to be?

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2. haha well I had a quick shot at it, and my answer feels wrong so I will post it and have someone explain....

I have the circumference of the earth in metres as 40,000,000
> diameter<-40000000/pi
> diameter
[1] 12732395
so the diameter of this earth is 12732395

I am adding 1 metre stick to each radius, and 2 to the diamter and times by pi to get the new circumference;
> (diameter+2)*pi
[1] 40000006

giving 6 extra metres, which seems crazy...

3. Ah thinking about it, each extra unit diameter gives me Pi units extra circumference, so the answer is 2 Pi metres extra?

4. Originally Posted by tolland
Ah thinking about it, each extra unit diameter gives me Pi units extra circumference, so the answer is 2 Pi metres extra?
Assumption: The Earth is a sphere

Let r be the Earth's radius such that $\displaystyle r = 4 \times 10^7m$

Let the stick be $\displaystyle \delta r$

Take a small bit of "width" of the Earth and let it tend to 0 so that it can be ignored giving a 2D model.

The original circumference of the circle is $\displaystyle C_r = 2\pi r = 8\pi \times 10^7$ whereas the new area of the circle would be $\displaystyle 2\pi (r+ \delta r) = 2\pi (4\times 10^7 + 1) = 8\pi \times 10^7 + 2\pi$

The difference would be $\displaystyle 2\pi \: m$

5. Excellent!

You know the strange thing is, it doesn't even matter what the radius of the great circle is. The rope will always be $\displaystyle 2\pi \ \ or \ \ 6.28$ units longer if the radius is increased by 1 unit.

$\displaystyle C=2\pi r$

$\displaystyle C=2\pi(r+1)$

$\displaystyle C=2\pi r + 2\pi$