The Great Railroad Track Prank




Once there was a small private railroad made up of a single straight track, exactly one mile (5280 feet) long. It was fastened to the ground at either end. One night, several practical jokers cut the track in the middle, and welded in an extra segment exactly 1 foot long. Since the ends of the track were fixed, this caused the track to bow upward into the air in an arc of a circle. (The length of this arc was 5281 feet.)




How high was the center of the arc above the ground? (You'll be surprised!)




HINTS: Draw the whole sector (wedge) of the circle; call the radius R. Let $2\theta $ be the central angle. Draw a radius from the center of the circle to the middle of the arc, and let $b=\frac{5281}{2}$ be half the arc. Note that $b=R\theta $ (so $R=b/\theta $). Let $2a$ be the chord of the whole arc, and note that $a=\frac{5280}{2}$. Find $\sin (\theta )$ in terms of $a$ and $R$, and hence find an equation involving $\theta $, $\sin (\theta )$, and $a$. Use Newton's method to solve this (for $\theta $). Once you've found $\theta $, find $R$, then use the equation for $\cos (\theta )$ in terms of $R$ and $R-h$ to find $h$.