Bending a long bar
A nice thinker from Futility Closet:
A rail one mile long is lying on the ground. If you push its ends closer together by a single foot, so that the distance between them is 5279 feet rather than 5280, how high an arc will the rail make?
Feel free to have a go yourself! Spoilers below the line.
Assuming the rail forms the arc of a circle, I’m not certain it’s possible to solve this analytically (it has
Setup
Suppose the track forms an arc of a circle, with a radius of
Then the length of the rail is
The distance between the ends of the rails is
So, working in feet 1, and calling
; and .
And we want
Solving
I think we can assume that
If we divide the two equations, we get
However,
We’re aiming for
How about
Finishing up
Now we’re getting somewhere!
All we need is to work out
For once, the imperial system’s fabled divisibility works in our favour: 5280 ÷ 24 is 220, and nine times that is 1980.
We still have to square root that…
Whoosh
“I might have known you’d show up once all the hard work was done.”
“1980 is 44 × 45, so its square root is 44.5, less
“How very Fibonacci.”
“Quite. So I’d estimate 44.49.”
“The Futilitarians just say ‘more than 44.’”
With something between a tut and a pshaw, the Mathematical Ninja was off.
Footnotes:
1. Ugh. Don’t tell the Ninja