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Putting a Ribbon around the Earth

Date: 12/23/2006 at 16:38:46
From: Tim
Subject: circumference of the earth 

If a ribbon around the circumference of the earth is 24900 miles, and
we wanted a ribbon to be 1 inch above the surface, what would be the
length of the extra ribbon we would need?

I was told all you had to do to find the answer is take the 1 inch
times 2 times 3.14 = 6.28 inches.  If this is correct then the same
length would add 1 inch the the circumference of a basketball.  Are
you starting to scratch your head, too??



Date: 12/23/2006 at 22:18:07
From: Doctor Rich
Subject: Re: circumference of the earth

Hello Tim,

Thank you for contacting the Math Forum.

Your question is a fairly well known one, because the results seem
hard to believe.  But it really is true.  Let's look at why. 

Assuming that the earth is a perfect sphere, we would have a circle
with a circumference of 24,900 miles at the equator.  That means the
radius would be 24,900/(2*pi) or about 3963 miles.

What if we think of the loose ribbon as a second circle around the
earth with a radius that is 1 inch longer than the radius of the 
earth? 

             
               Ribbon
               *     *
         *     *     *     *
       *  *               *  *
       
     * *        Earth        * *
      
                       r      + 1 inch
    * *           x-----------*-*
    
    
     * *                     * *
       
       *  *               *  *
         *    *      *     *
              *      * 


The circumference of a circle C with radius (r) is 

    C = 2 Pi(r)

The circumference of a circle C with radius (r + 1) is 

    C = 2 Pi(r+1)
    C = 2 Pi(r) + 2 Pi 

From this you can see that the circumference will increase by 2Pi.  In 
this problem, since the radius increases by 1 inch, then your answer 
of a 6.28 inch increase in the length of the ribbon is correct. 

Note that no matter what the original radius is (the radius of the
earth or the radius of a basketball), increasing it by 1 inch will
always lead to an increase of 2Pi or 6.28 inches in the circumference.

I think part of what's so hard to believe about this problem is that
when we think about the circumference of the earth, we assume it has
to be some really large number, especially in comparison to the
basketball.  But as the math shows, you'll get the same answer for any
size original radius!

I hope that helps solve your dilemma.  Please contact me if I can be 
of further assistance.

- Doctor Rich, The Math Forum
  http://mathforum.org/dr.math/ 
Associated Topics:
High School Conic Sections/Circles

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