A railroad one mile long is anchored at both ends. On a hot day it expands one foot and buckles. Approximately how high off the ground is it at its midpoint?

[Editor's Notes: (1) You can assume the buckled rail has the shape of an arc of a circle. Books on Strength of Materials would assume a parabolic shape. (2) The problem asks for an "approximate" answer; how "approximate" is your answer? Try to get upper and lower bounds for the true height.]

Hint 1

Draw a picture. Include the radii of the arc.

Hint 2

Here's a picture:
railroad track
We've used letters here for all the measurements; we already know that 2c = 5281 feet and 2d = 5280 feet. If we knew the length k we could calculate the height h using the Pythagorean theorem, but we don't know k. Try to get an upper bound for k (this is easy), then get an upper bound for h. Then try to get a lower bound for k (this is a lot harder) and use that to get a lower bound for h.

Hint 3

The easy upper bound for k is c (k is a straight line, so its length is less than any other curve between the same endpoints). Therefore we get an upper bound for h as

This seems much too big; imagine that a mere increase in length of 1 foot over a 1 mile stretch would raise the midpoint 51 feet! Of course, our figure is an upper bound, so the true value might be a lot smaller. We'll investigate a lower bound next.

The next three hints give a way to get a lower bound for k.

Hint 4

Show that k > (d + c)/2 and use this to get the lower bound for h.

Hint 5

Write the lengths in terms of r and the angle a. We can get these from the definitions of sine, angle (in terms of arc length), and by bisecting the angle a to calculate k/2. We get

length in terms of r and a

so that we want to show

inequality for a and beta

Hint 6

Draw a sector of the unit circle. We will prove the inequality by using the three areas in the figure:

figure for inequality

From the figure we can calculate the areas A and A+B (using area = (1/2) base times height) and A+B+C (using area = (1/2) arc length):

areas of A, A+B, A+B+C

From the figure, the area B is half the area of the rectangle abcd, and the sliver C is part of the other half and so is less that B. This gives us the inequalities:

proof of inequality

which is what we wanted to prove. Now we can find a lower bound for h

The Rest of the Solution

The lower bound is

lower bound calculation

So we know the true height at the midpoint really is somewhere between 36.33 feet and 51.39 feet, all from a little 1-foot increase in the total length!

Observe how sensitive the answer is to little uncertainties in the length k; we know k is between 2640.25 and 2640.5, a tolerance of 1/4 foot, but this causes an uncertainty in the height of about 15 feet!

A More Precise Estimate

Another way to work this problem is to figure out the angle a. By dividing the first two equations in Hint 5 we cancel the unknown r and we get an equation

Unfortunately there is no explicit solution for this equation, so you have to get a numeric approximation. You can do this with calculus, or you can solve it with a graphing calculator such as the TI-83 Plus. Then you can figure r and then h. The Reference works this out in detail, and gets 0.033708 radians for the angle and 44.499 feet for the height.

References

Forman S. Acton, Numerical Methods That Work, Mathematical Association of America, 1990, pp. 3-4, 67-69. Works out a more precise estimate. When this book was written in 1970, the main tools were a slide rule and a set of trig tables, and it was challenging to calculate all these numbers to an adequate precision. Acton takes care to arrange the calculation so that cancellation will not rob us of our significant digits. Now that we have 10-digit calculators and desktop computers, the brute-force method is adequate for this problem, but Acton's methods are still valuable for many other problems.
Click here to view the original problem submitted by Miran.