How much will a band around the equator rise above the surface if it is made one meter longer?

Additional material

[ Editor: a variant, rec.puzzles, March 1995. ]

In article <D5G5I3.Drn@Newbridge.COM> wprager@Newbridge.COM (Walter
Prager) writes:
>Remember the puzzle posted a short while ago about adding a foot to a
>length of rope encircling the Earth at the Equator?  Well, I showed it
>to my personal puzzles group and I got an interesting variation.  I
>originally asked them what is the maximum height that the rope reaches
>above the Earth if you added 1 foot to it and kept it as a perfect
>circle.  The answer I got was 1/Pi, if you allow the rope to touch
>the Earth in one spot.
>
>That got me to thinking: what is the maximum height the rope can reach
>if you forget about keeping it as a circle?  In other words, grab the
>rope in one place and pull hard.  How high will it go?
>(I guess you'll need the radius of the Earth in this case, so I'll take
>someone else's answer from before--7564 km, or appr. 24816273 feet.

[ Editor: the Earth's equatorial radius is about 6378 km (about 20.9
million feet), not 7564 km. ]

Assuming the equator is a perfect circle, and the rope will lay flat
along the equator where possible?

I get approximately 191.1 feet from the ground; it will lay flat across
the ground for all but the last 18.45 miles in either direction from
your current position.

There may be an easier way to get the right number, but I used
successive approximations for the angle between that point 18.45 miles
away and the present point on the ground, and calculated the size of the
right triangle created between the point 191.1 feet above the ground,
the tangent point 18.45 miles away, and the center of the earth.

-- Doug Nelson <nelson@msu.edu>