I think we can assume we are dealing with a perfect sphere. The object of the question is to see if you add 10 meters to the wire, how much space will be under the larger circle. So if you have a wire tracing the circumference of the sphere, 40,000 kilometers long, it is forming a circle. The radius of a circle is r = circumference / (2*pi)
r = 40,000km / (2*3.1415926535)
This gives the circle a radius of 6366.1977239 km or 636619.77 meters
If you add 10 meters to the wire and wrap that around the same perfect sphere
radius = 40,000.01km / (2*3.1415926535)
This gives a radius of 6366.1993154 km or 6366199.93 meters
So with 40,000 km of wire, the radius is 636619.77 meters
So with 40,000.01 km of wire, the radius is 636619.93 meters.
The difference is .16 meters, or .52 feet. So there would only be about 6 inches clearance to walk through.
To avoid retracing all your steps it should suffice to say that "delta" 10 in circumference will result in "delta" 1.6 in radius. You remember "delta", it means "change".
The equation is:
C=2*Pi*R therefore for every 10 units of C (circumference) you would need to feed the equation 1.6 units of R (radius) to equalize both sides.
Now, you gotta figure out where you made the ten-fold mistake. You got 0.16 in the place of 1.6.
Oh, I see where the mistake took place. It's not 636619.77 meters it is supposed to be 6366197.7 meters.