[superprismatic]

Suppose the earth were a perfectly smooth

sphere with radius 6,371,000 meters. Now,

there's an old problem which supposes that

a rope girdles the earth along a great

circle. An extra meter of rope is then

spliced into it. You then assume that the

rope is raised the same amount above the

earth's surface all the way around. You

are asked to determine what the amount of

the rise is. It's pretty easy to see that

the rise is independent of the radius of

the earth and is 1/(2π) meters. But we

consider a different problem here:

Suppose the earth were a perfectly smooth

sphere with radius 6,371,000 meters. Now,

suppose a rope girdles the earth along a

great circle. An extra meter of rope is

then spliced into it. Now, grab the rope

in one spot and pull it taut directly away

from the center of the earth. How far

above the surface of the earth will it be

able to be pulled? Give the answer to the

nearest meter.

[Peekay]

Let

r be the radious of tha sphere.

y be the distance from the ground to the top of where you pull the rope

a be the angle from where you stand to where the rope is tangent to the sphere

x be the distance from where the rope is tangent to the earth to where you are pulling the rope

tan (a)=x/r

hence

x = r*tan(a)

2*pi*r - 2*a*r + 2x = 2*pi*(r+1)

substitute x

2*pi*r - 2*a*r + 2*r*tan(a) = 2*pi*(r+1)

2*r*tan(a)-2*a*r=2*pi

a~~0.6528°

We also know:

r^2 + x^2 = (r+y)^2

substitute x = r*tan(a)

r^2 + (r*tan(a))^2 = (r+y)^2

solve for y

y ~ 413 m

In the following step:

2*pi*r - 2*a*r + 2x = 2*pi*(r+1)

How are you getting the (r+1)? 1 meter was added to the circumference, not the radius.

[superprismatic]

I'm glad this little puzzle tickled as much

interest that it did! It really seems counter

intuitive. Congratulations to EventHorizon

for his quick correct response. For a problem

where accuracy is paramount, he got the height

right on the nose. Thank you to everyone who

worked on it.

[Guest]

A funny application of this new found phenomena: When people pull the waist-line of their pants out to show how much weight they've lost, you can show them the math and convince them they didn't really lose much circumference.

edit: Just don't start with, "assume the radius of your waist is that of the earth..." as it might make them feel bad.

Wow! I nearly spewed coffee all over my desk. Thank's for the laugh; I've got to put that in my signature.

[Peekay]

I'm glad this little puzzle tickled as much

interest that it did! It really seems counter

intuitive. Congratulations to EventHorizon

for his quick correct response. For a problem

where accuracy is paramount, he got the height

right on the nose. Thank you to everyone who

worked on it.

Thanks Superprismatic for a wonderful puzzle!