Brain Teaser, string around the world?

Question

Assume the earth is a perfect sphere, with the surface at mean sea level. Now wrap a string around the world on the surface at the equator.  Now assume you take that same string and add 10ft onto it, then some how  magicallly elevate it off the surface of the earth an equal distance off the ground all the way around the earth.  The circles the string made, and now make are concentric circles in the same plane.  The actual length of the string is not affected by temperature or any other factors.  Now here is the question.. How far off the ground will the string be?  Is it so close that it is hard to measure (i.e. like something less then an millimeter), or could a small dog walk under it, or could a human walk under it?

sofarsogood · Accepted Answer

Call radius R feet, so circumference is 2pi*R. Add 10 feet, then 

2pi*R + 10 = 2pi(R + x), where x is the added height.

2piR + 10 = 2piR + 2pi*x
10 = 2pi*x
x = 10/2pi = 5/pi is between 1 and 2 feet. Small dog, or very good limboer.

sft2hrdtco · Answer

Say the radius of the Earth is "x" (in units of feet), this means the Earth's circumference is equal to: 2*pi*x = C

Adding 10 feet to this means the length of the new rope is 2*pi*x + 10 = L

The radius of this new circle can be defined as (x+y) where "y" is the distance off the ground.

We know that the length of the longer rope minus the circumference of the earth is 10, so:

2*pi*(x+y) - 2*pi*x = 10
2*pi*x + 2*pi*y - 2*pi*x = 10
2*pi*y = 10
y = 5 / pi

Because "y" the distance from the ground of the Earth to the top of the rope has no extra variables it, it shows that it is completely independent of the value of "x" or the radius of the Earth.

y = 5ft/pi ~ 1.59ft

Anonymous · Answer

A foot and a half.  Actually, ten feet divided by two pi.

lightningviper · Answer

to close to call,

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Brain Teaser, string around the world?

4 Answers