9514 1404 393
Answer:
about 44.5 feet
Step-by-step explanation:
We can write relations for the height of the rail as a function of initial length and expanded length, but the solution cannot be found algebraically. A graphical solution or iterative solution is possible.
Referring to the figure in the second attachment, we can write a relation between the angle value α and the height of the circular arc as ...
h = c·tan(α) . . . . . . where c = half the initial rail length
Then the length of the expanded rail is ...
s = r(2α) = (c/sin(2α)(2α) . . . . . . where s = half the expanded rail length
Rearranging this last equation, we have ...
sin(2α)/(2α) = c/s
It is this equation that must be solved iteratively. We find the solution to be ...
α ≈ 0.0168538794049 radians
So, the height of the circular arc is ...
h = 2640.5·tan(0.0168538794049) ≈ 44.4984550191 . . . feet
The rail will bow upward by about 44.5 feet.
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<em>Additional comments</em>
Note that s and c in the diagram are half the lengths of the arc and the chord, respectively. The ratio of half-lengths is the same as the ratio of full lengths: c/s = 2640/2640.5 = 5280/5281.
We don't know the precise shape the arc will take, but we suspect is is not a circular arc. It seems likely to be a catenary, or something similar.
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We used Newton's method iteration to refine the estimate of the angle from that shown on the graph. The iterator used is x' = x -f(x)/f'(x), where x' is the next guess based on the previous guess of x. Only a few iterations are required obtain an angle value to full calculator precision.
