Loosening the Earth's Belt

Date: 1/18/96 at 23:44:34
From: Anonymous
Subject: Algebra II

The Earth's Belt problem.

The earth has a belt around the equator that is perfectly fit. 
If a measurement of 1 foot is added to the belt, theoretically,
the belt will no longer touch the earth at any point. What is the
new distance between the surface of the earth and the belt.

Using the encyclopedia, I found the radius of the equator. 
Radius = 3963 miles, Circumference = 24900.26 miles. The answer 
I got was phenomenal. It was .16 feet. I expected it to be 
something like .000385 feet (at least 3 zero's before any other 
number). The technique I used was convert the miles into feet, 
and find the distance between the radius of the old belt and 
the new one.

Date: 1/21/96 at 21:14:6
From: Doctor Ken
Subject: Re: Algebra II

Hello!

Yeah, a lot of people are surprised when they do this problem.  By 
the way, the way the problem is usually asked is "how much extra 
belt material will it take to raise the belt 1 foot off the 
Earth's surface?" or something along those lines.  But it's quite 
astonishing (until you realize why) that a very small amount of 
new belt material will raise the belt pretty far.

Here's why.  Instead of writing the radius of the Earth as a 
number, let's just call it r.  So the old belt will be 2*Pi*r feet 
long, and the new belt will be 2*Pi*r + 1 feet long.  You want to 
find out "for what new radius, call it R, will 2*Pi*r + 1 = 
2*Pi*R?  Solving that equation, we get R = r + 1/(2*Pi).  So no 
matter what the radius of the Earth is, the new belt will hover 
off the ground at a distance of 1/(2*Pi), which is about 0.16 feet.

-Doctor Ken,  The Math Forum