# King Kong and the Railroad Track problem.

• Oct 27th 2010, 04:53 PM
henryhighstudent
King Kong and the Railroad Track problem.
Hi, I'm taking AP calculus in high school and I am completely baffled on this one.. (Surprised)

Suppose you have a straight and level 1-mile long railroad track. Imagine that one end of the track is fastened to the ground and the rest of it is not fastened. King Kong walks by the unattached end and pushes it exactly 1 inch towards the fastened end so that the track bows up in a circular arc. How high is the track above the ground in the middle of the circular arc?

So, the track is basically in an arc since King Kong pushed it an inch inward.
• Oct 27th 2010, 06:13 PM
Ackbeet
So it seems to me that the main issue here is that, considering the circular arc, you don't know what angle the arc describes. But you do know the length of the chord corresponding to the arc, and you know the arc length. Let me outline one method of solution.

1. You know that the chord length is given by $\text{cord length}=2r\sin(\theta/2).$

2. Since $\text{arc length}=r\theta,$ you now have two equations in two unknowns, solving for $r$ and $\theta.$

3. Call the desired distance $h,$ and the perpendicular from the center of the circle to the chord has length $\ell.$ Then $h+\ell=r.$

4. From basic trigonometry, you also know that $\dfrac{\ell}{r}=\cos(\theta/2).$ From this you can get $\ell.$

5. Since $h=r-\ell,$ you are now done.

Make sense?