Ring Around the Earth

How much material to extend a railroad around the Earth?

Imagine that in a future era humans decide to build a high speed train circumnavigating the globe at the equator. Ignore the impracticalities of such an undertaking (e.g. building railroad tracks across an ocean) and think about this question: How long does the track need to be? This is easy to answer if you know the circumference of the Earth, which is 24,902 (or roughly 25,000) miles at the equator.

Now, imagine our engineering team determines that, for technical reasons, the track needs to be elevated two feet off the ground. We were already planning to acquire 25,000 miles of track so the question is this: how much additional track do we need in order to build our equatorial railroad two feet above the Earth’s surface?

Leave your answer in a comment below. I’ll post a solution and the names of all puzzle solvers on Monday.

Solution: Think of the equator as a giant circle. Let’s call the radius of that circle R, the distance from the center of the Earth to any point on the equator. The elevated track can also be thought of as a giant circle, one with radius R+2 (since the track is two feet off the ground). Now the question of how many additional feet of track we need boils down to subtracting the circumference of the elevated track from the circumference of the equator.

The circumference of any circle is given by 2*Pi*r, where r is the circle’s radius so with a bit of algebra we get:

more track needed = track circumference – equator circumference = 2Pi(R+2) – 2*Pi*R = 2Pi(R+2-R) = 2Pi(2) = 4*Pi ~= 12 So, to raise the track by two feet all 25,000 miles around the Earth, you only need to add approximately 12 feet of track! Even more amazing, because the solution is independent of the starting radius, the same answer applies to every possible circle. For example, if you tied a string around a basketball and then decided to raise that string two feet above the surface of the basketball, you’d need precisely the same amount of additional string: 12 feet.

This is a problem I first heard as a kid and to this day it still strikes me as incredibly surprising. Hat’s off to Mudassir Ansari, Dan Stoops, John Baldi, Ricardo Agudo and Jim Goss for coming up with the correct answer!