GeekDad Puzzle of the Week: All Tied Up – Solution

Thanks to all who submitted solutions to this week’s puzzle. (Yes, this was an easy one, but don’t expect such things forever!) Congratulations to Erik DeSimone who receives a $50 coupon code to ThinkGeek! Solution after the jump.

Special thanks to ThinkGeek for providing our prizes!

One quality that separates you from all dads is that you consider no project too large as long as it is for your kids. Your youngest has come to you and asked how much rope will wrap around the Earth.
Not content to simply answer the question, and being a right and proper geekdad, you endeavor to tie an actual rope around the Earth’s equator.
You head to your local building supply superstores (three of them actually) and clear them out of rope. As you’re setting up for your project, your neighbor, being the boring and negative type, challenges you to wrap the rope around the Earth’s equator elevated from the ground by one yard at all points along the rope.

How much rope will you need? (Assume the Earth is perfectly spherical.)

SOLUTION

~43,826,323 yards of rope. If you used Wikipedia for the Earth’s diameter, your solution should be within shooting distance of this amount. The key to knowing how much rope (which could vary depending on which equatorial measurement you used) is knowing that the amount of extra rope you need to wrap the rope 1 yard off the ground is the same no matter what.

The circumference of a circle is 2?r, where r is the radius of the circle in yards. If you want a rope that is one yard above the ground, the radius is larger by one yard. So the new radius is r + 1.

Let x be the amount of extra rope required. So:

     x = (2?(r + 1)) – (2?r)
     x = (2?r) + (2?) – (2?r)
     x = 2?

So x is about 6.2832 or 6.3 yards. Add this to your circumference calculation and you should arrive at an amount that is close to the above or near ~43,828,577 if you’ve used other measurements. Note that this answer does not depend on the radius of the sphere. If you were attempting to tie up a tennis ball rather than the Earth, the amount of additional required rope would be the same.

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