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A 100 meter long train rail expanded under the sun. The center of the rail went up 1 meter above the group while the two ends are still attached to the ground - The whole rail forms an arc.

How long is the arc?

note: you can assume the ground is flat and the rail was straight on the ground and ignore the depth of the rail.

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  • $\begingroup$ FYI: This question appeared on a mathematics contest I participated in many years ago. The test contains 20 interesting but vert hard questions involves probabilities, logic, geometry and numbers, and I would consider each question as a puzzle. I was not able to solve the problem at that time so I thought it would be a fun question to ask since it's quite puzzling. If you find this question is out of topic, then I do apologize as my interpretation of "puzzles" may be different than yours. But I will not ask any math problem here any more. $\endgroup$ – Manto Jul 3 '18 at 8:12
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You could have at least sandwiched it in a riddle...

Imagine the first length of the rail as a straight line - the chord of a circle.
enter image description hereDraw two radius lines perpendicular to the expanded rail - one from the end, one from the middle of the rail, making one half of the expanded rail and the two drawn radii a section of a circle.

The radius line that comes from the middle of the expanded rail bisects the chord (line where the unexpanded rail was) and is perpendicular to it.

Thus the full radius, the middle radius (this side is 1m shorter), along with half of the original length of the rail (50m) form a right triangle.

Using the Pythagorean formula, we can then find the length of the radius:
$$50^2 + (r-1)^2 = r^2 $$
$$50^2 + r^2 - 2r + 1 = r^2$$
$$50^2 - 2r + 1 = 0 $$
$$ 2501/2 = r $$
$$ r = 1250.5 $$
We can then use trig to find the included angle:
$$sin(\theta) = 50/r = 50/1250.5$$
$$\theta = 0.0399946679 rad$$
The length of half of the expanded track, then, is:
$$r*\theta = 1250.5*0.04.... = 50.0133323m$$
Thus, the total length of the track is 100.026665 m

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    $\begingroup$ Please do not answer this kind of questions, we are not do-my-maths-homework-for-me.stackexchange.com, and we don’t want to encourage anyone trying to use us as such. $\endgroup$ – Bass Jun 30 '18 at 20:19
  • $\begingroup$ @IronEagle, thanks for solving the problem. I've been a software engineer for so many years and haven't touched math puzzles until I saw that math contest I did years ago when I was cleaning my storage. $\endgroup$ – Manto Jul 3 '18 at 7:58
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    $\begingroup$ @Bass, I am a new user to Puzzling and may not know all the rules and lines between a puzzling problem from math contest or a "Riddle" that comforts the majority's definition of "Puzzling". If I made a mistake, you can say it nicely and educate me so I'll try to contribute a more appropriate question next time. I find your comment not only mean, but also very discouraging, especially to new users like me. I am not asking a problem from my "homework", I am asking a nostalgic math contest problem that I did many years ago and has been puzzling me since $\endgroup$ – Manto Jul 3 '18 at 8:08
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    $\begingroup$ @Manto, very sorry about that. All the tell-tale signs of this being a homework assignment were there, and I lost one version of the comment to an accidental page refresh already, so the final version certainly came up too terse. To better get accustomed to the site's customs, you might want to start by taking the tour. Again, very sorry to have offended you, and I'm looking forward to seeing more questions and answers from you. $\endgroup$ – Bass Jul 3 '18 at 12:14

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