That fact that the answer is the same for both, yes, but that you only need to add 6+ feet to bring the earths diameter a foot off the surface seems incredible at first blush.
The band around the earth problem reminds me a bit of the rail problem, where a simple geometric construction has a nonintuitive result:
There are two steel rails laid end-to-end, each a mile long. A flexible joint connects them at the center of the two-mile long stretch. The far end of each rail is pinned to the ground. Then each rail expands by one inch but remains straight, causing the center joint to rise off the ground. How high does it rise?
A simple application of the Pythagorean theorem shows that the joint rises almost 30 feet off the ground.
Also the question did not clarify that the cities were on Earth
So the expansion results in a right triangle?
Well, two right triangles.
I guess I’d have to see it.
One right triangle to the left, with base 5,280ft, height 30ft, hypotenuse 5,280ft+1in, one triangle to the right, with base 5,280ft, height 30ft, hypotenuse 5,280ft+1in.
Any isosceles triangle can be divided into two right triangles.
DOH! is a complete sentence.
Wow! That does seem incredible.
Putting the problem into inches makes the math more understandable I think.
one mile of track in inches, a = 63360
one mile of track plus one inch in inches, c = 63361
a2 = 4,014,489,600
c2 = 4,014,616,321
a2 + b2 = c2
b2 = c2 - a2
4,014,616,321 - 4,014,616,321 = 126721
√126721 = 355.98 or ~30 feet
The size of the squares are so large that I think we can’t comprehend how huge they are. The difference between them is tiny – .003% – but even a tiny percentage of a huge number is bigger than we realize.
I doublechecked my numbers, and they work out, but triplecheck me anyway.
This is a reverse application of the fact that a tiny amount of force sideways exerts a massive amount of force pulling on a rope. Moving the end a tiny amount with a force effectively only limited by the expansion of the rope.
Here’s an interesting fact I was recently reminded about.
Charles Lindbergh, when flying the Spirit of St Louis from the U.S. to France to become the first person to fly a solo transatlantic flight, had no windshield.
He had a periscope, and windows to his sides, but otherwise his view was just of his instruments.
Then again… What’s directly in front of you in the air that you need to see?
Here’s another interesting one when it comes to math and geometry:
Let’s say you want to measure the length of the shoreline (perimeter) of a small island. How are you going to do it? What length resolution are you going to use? That’s important, because you will get wildly different answers depending on what you use. Let’s say you measure it using a yardstick (length resolution of 36 inches), and it shows the shoreline is 3.2 miles long. If you instead used a resolution of 1 inch, the shoreline might be 100 miles. If you were to use a resolution of 1 millimeter, the shoreline might be a million miles. In other words, as your measurement becomes more “accurate” and “precise,” the distance increases exponentially, resulting in ridiculous numbers. The same is not true when measuring area. I’ve always found this fascinating.
For purposes of beachfront property, docks, moorage, etc. what is the practical measurement baseline? Something like rods, chains, or furlongs?
When landing, it’s useful to see in front of you.
Today I learned:
The characteristic holes in Swiss cheese (technically Emmentaler) have been shrinking and disappearing over the last couple of decades. After investigation, scientists discovered the reason — modern dairy systems produce milk that is too clean.
Not to worry! They have a solution. They’re just seeking approval from the authorities.
Nowadays, definitely. But Lindberg’s landing spot was probably just a big, completely-empty field. No other airplanes, no terminal buildings, no traffic-control tower, etc.