It's curious how people think about problems ... what they think is small and what they think is large. >Huh? Okay, here's a problem: A railway track is a mile long. Well, let's say 1000 metres, like so It is one continous ribbon of steel, pinned down at both ends, but free to move in between. The temperature rises and the track expands to 1002 metres. If the track buckles (as shown), how high would you think the centre rises?

>You expect me to solve that problem?
No, I expect you to guess at the answer.

>Uh ... I'd say something small ... like maybe a centimetre.

Okay, let's estimate that height. Assuming a right triangle, we have that h2 = 5012 - 5002 = 1001.     So our estimate is h = √1001 = 31.6 metres.

>You're kidding, right? A right-angled triangle? Are you kidding me?
Well, it's just a simple estimate ... but, as I recall from the actual solution to this problem, it's within 10%.

>So it buckles by roughly 30 metres?
Roughly. That's' close to 100 feet.
Interesting, eh? Most people would guess as you have. Something small. when it's actually HUGE!

>I think I'll avoid trains in hot weather.
That's not the way they build tracks. The sections are maybe 20 metres long, with expansion gaps to accommodate the expansion.
When the train passes over these gaps, you hear ...
>Clickety clack!
Sounds good to me.

Here's another one:

• An elastic rope is tied tightly about a spherical earth, like so:
  
• Then the rope is lifted above the surface by 1 metre, like so:
  
• How much longer must the rope be to encircle the earth in this way?

>Uh ... how big is the earth?
The circumference, hence the original length of rope, is about 40 million metres.

>Then I'd say several hundred metres .. at least. Maybe a thousand metres ... but I'm doing that estimate thing!

Okay, if the radius of the earth is R, the original length is 2πR metres.
Now increase the radius of the rope by 1 metre. It's length becomes 2π(R+1) metres.
The increase is then 2π metres or about 20 feet.

>Huh?
Interesting, eh?
For the first problem, people usually guess small when the answer is large.
For the second problem, they usually guess large when the answer is small.
Maybe they're thinking percentages.
A 1 metre expansion in 1000 metres? That's a small percentage, eh?
A percentage of a 40 million metre rope, extended to increase its stretch about one HUGE earth ... that must be large.
Whatchya think?
>zzzZZZ