Band around the Earth Puzzle - Solution
The Puzzle:
You come along at night, as a practical joke, and add just 10 meters to its length (one hundredth of one kilometer !)
It is now one four-millionth longer, and sits magically just above the ground at all locations
How far has it risen ... could a flea, a rabbit or even a man squeeze underneath it?
Our Solution:Use the formula Circumference = 2 × pi × Radius
Before: Original Circumference = 2 × pi × R
After: Original Circumference + 10m = 2 × pi × (R + Gap)
Subtracting the two:
10m = 2 × pi × Gap
So, the Gap = 10m / (2 × pi) = 1.6m approximately
So a man could fit under it easily (though he might bump his head)
Note: Adding 10m to the circumference of ANY circle increases the radius by 10m / (2 × pi), no matter what the original circumference was.
Before: Original Circumference = 2 × pi × R
After: Original Circumference + 10m = 2 × pi × (R + Gap)
Subtracting the two:
10m = 2 × pi × Gap
So, the Gap = 10m / (2 × pi) = 1.6m approximately
So a man could fit under it easily (though he might bump his head)
Note: Adding 10m to the circumference of ANY circle increases the radius by 10m / (2 × pi), no matter what the original circumference was.