You can actually work out roughly how far away the horizon should be surprisingly easily. Just use the fact that if R is the radius of the Earth and you are at height h, then from you to the horizon to the center of the Earth back to you is a right angled triangle with one side of length R and the hypotenuse of length R+h. Therefore the distance to the horizon is sqrt(2Rh+h^2) which is roughly sqrt(2Rh).
The Earth is roughly 6370 km which is not far from 6400, so if your eyes are 2m = 0.002 km up then the horizon is about sqrt(264000.002) = sqrt(25.8) km away, which is roughly 5 km or a bit over 3 miles.
If you apply this to a 6 km tall mountain, the horizon is about sqrt(263706) km away which is about 276.5 km. So something of the same height at the opposite end of the horizon would be 553 km away. So the top distance of 538 km is pretty close to the maximum that we would expect.
What about an airplane? An airplane flies about 11 km up. So it can see around 375 km. If you work it out, that puts the horizon about 3.37 degrees below horizontal. This isn't much, but if you take a plumb line and a right angle on an commercial flight, it is enough to actually see that the horizon is below horizontal.
I had this question worded differently asked to me when I interviewed to do an undergrad at Cambridge. If you have a rope that is wrapped around earth and you lift it off the ground as much as possible, and you see that it is 10km above the ground, then how long is the rope?
I was not able to answer that question right away, and their hint to draw it helped a lot. I did not end up getting accepted.
In this case the "arbitrary filter" was filtering applicants to a university for very basic math knowledge, so I think assuming that it's correlated with success at that university is not very far fetched.
I don't think after playing 20 or so levels I was much smarter for it. It turned out to be a bunch of rules that were extremely limited. You do begin to appreciate trig much more.
If you are being given a geometry subject test and are prepared for it, this is fair game. If the question is just given out of the blue as a brain teaser, then no, the university is not mining useful signals at all.
When I did my university entrance interview (for maths, at Cambridge) the interviewers were clear that they expected that some subset of candidates would have seen the problem before, and some wouldn't -- for those in the first set they'd get them to quickly go through the problem and move onto the later parts which would be new to them; for those in the second set they'd provide sufficient guidance to let the candidate walk through the problem. The point was to get any particular candidate to a point in the problem sequence where this was something new to them, and then see how they tackled things. The idea that some applicants (usually from public schools) would have been very highly prepped for interview and others (usually from state schools) would not was clearly something they were well aware of and setting their interview design up to handle.
But how do you know it's a new problem? You can always fake a little struggle and thinking your way to the solution for a problem that you know the answer already.
I don't see anything tricky about that problem. It requires some junior high geometry and high school trig, and yes, I suppose you need to know the radius of the earth offhand. Still. That strikes me a lot more as a math fizzbuzz than as a brain teaser. A college applicant can be reasonably expected to know this stuff.
I suppose you need to know the radius of the earth offhand.
Or you can easily derive the radius if you remember that a meter was initially defined as 1/10,000,000 of the distance from the equator to the North Pole. So you can easily calculate the Earth's circumference, and from there the radius: 10,000,000 * 4 / 1000 / 3.1416 / 2 = 6366 km.
It also seems to be a less coherent version of the chicken from Minsk puzzle.
Perhaps it's multiple choice.
A. Dreadfully sorry, but I'm afraid I haven't memorized the circumference of Earth.
B. What do you mean? Over the poles or 'round the equator?
C. Is that 10 km above at all points on the rope, or just one?
D. What's the rope made of? Jute? Sisal? Nylon? How far apart are the supports? Young's modulus...
E. I don't know about that, but I have inherited a rather large quantity of string in 3-inch lengths...
There is a lot more than the Pythagorean theorem in that question. It would require remember a lot of middle school theorems from a long time ago; frankly calculus would be better.
The version I got, in a tech company interview, was how much longer would you need to make the rope to allow a cat to go under it. They were basically just look for 2 x Pi x Cat in this case.
That question is ambiguous. If you only need a single cat to go under the rope at some point, you just add a small arch for the cat to walk through. This requires a length of rope somewhat less than twice the height of the cat.
2 x Pi x Cat would levitate the rope by the height of the cat along the entire circumference of the Earth, allowing a billion cats to go under it simultaneously.
You could even make do with no additional rope if you're allowed to take advantage of terrain.
The answer to the first interpretation is considerably less than twice the height of the cat, and I think it's an interesting result as well:
Assuming a 0.3 meter high cat, you only need ~0.12mm of extra rope (about the width of two sheets of paper) to be able to pull it up at a point and let the cat through.
Follow-up question - after extending the rope by 0.12mm and pulling it up to allow the cat to pass under it, how far away are each of the two points where the rope lifts off the ground?
Haven't done any calculations but I would guess half of the Earth circumference? because if you want to minimize the length of the rope then you have to maximize distance between those two points.
If kovek's question was like the first image there, it'd be a simple radius/circumference calculation (2 * pi * 5km). But it sounds like the second image to me - which makes it into more of a horizon line problem.
Yeah, I know that. I'm just trying to add to the history of rope/earth/cat interview questions here. This was ~15 years ago by the way and to be clear we're talking about cats being able to pass anywhere, not the line of sight problem, which I agree makes it a little more interesting..,
I'm not sure if I understood you, but 2pi(height of the cat) would allow to make two additional round loops, each sufficient for a very cylindrical cat to go through.
The "multiply by 2pi" version of this problem is the one where you have to lift the rope by a cat's height all around the entire earth. So if you have an army of southern-hemisphere cats and they need to get to the northern hemisphere, but you have cunningly put a rope around the equator because you don't want dogs to invade, how much extra rope do you need to lift it to just let in the cats but not the dogs?
(I'm assuming all cats are shorter than all dogs.)
To be clear, they gave you the radius of the earth here, if you didn't already know it by heart? Otherwise, I'm not getting how you could solve this problem (assuming they didn't give you the angle that the arch formed, for example).
You don't need to know the radius of the earth unless you're asked for a numeric answer. The rope is 2 * pi * 10km longer than the Earth's circumference, since its radius is 10km more.
I think this is the wrong interpretation; your answer would be correct if the rope were hovering at a height of 10 km along the full length.
My interpretation is that the rope is pulled taut to a height of 10 km at a single point, so that it runs in a straight line to the horizon in either direction and lays on the ground the rest of the way; this is also relevant to the question of longest line of sight from a given height.
Yes, but I only posted that so you'd see which problem the GP is referring to: the second one. Unless somehow the rope is some special kind of non-flexible rope that always assumes the shape of a circle.
fwiw, this method doesn't seem to predict the list very well
as its not a list of just the highest peaks in the world
longest sightlines also depend on the surrounding topography that blocks or doesn't block a sightline
this seems to account for the numerous Spanish entries on the list, where the large Spanish plain with high peaks on either side, enables you to see from 1 peak to the other, w/o anything on the plain blocking your view
It didn't claim to predict the list very well. Its purpose is to give intuition about how far around the Earth we can see.
Also it shows that simply being tall (airplanes go higher than Everest) doesn't get you on the list. The second paragraph puts an upper bound on sight lines between two mountains of a given size. The longest sight lines are pretty close to that limit. And it gives a simple experiment that anyone can carry out from an airplane which will let you see the curvature of the Earth.
This is too simplistic of an approach. Suppose there is another 6km tall mountain just beyond the horizon, it will have a farther line of site to the peak of it than to the horizon just in front of it. This list seems to be a much more rigorous attempt.
It is meant to show how to calculate the position of the horizon on a smooth planet. Obviously your line of sight typically includes tall things beyond the horizon. However this calculation lets you know both how far away your horizon is, and how far beyond that a tall object could be seen if nothing else was in the way. Which puts an upper limit on how far away it can be.
A little bit of elementary geometry gives you a lot more than I expected it would the first time I amused myself by figuring the calculation.
Actually finding those points requires a lot more work, which this list does.
Tell him to watch a f'in sunset.
Or the eclipse on Aug 21, which will happen exactly to the second predicted by astronomers.
Or look through binoculars at a ship sailing away over the ocean.
Or look at the stars rotating around the north/south pole at night.
Or fly to Australia and measure the mileage between any two cities and ask him which map is more accurate. Make sure to have a return flight through South America and ask him how long the flight took.
Or ask him why low tide is always when the moon is at the horizon.
Or ask him what causes phases of the moon.
Or ask him why the azimuth of Polaris is always equal to the observer's latitude on Earth.
Or ask him where satellites are.
All of these strategies are incorrect, as they presume that logic, mathematics, and rationality will have any impact on one who clearly places no value on empirical evidence whatsoever.
What you do is put up $10000 of your money against $1000 of his (10 to 1! You can't lose!), on a bet where the geographical mathematics ensures that you are 100% guaranteed to win. Or something--anything--that can demonstrate with real tangible consequences how being intentionally stupid can hurt you and those around you.
Build two identical UAVs, and run a race between two distant cities. One UAV follows a great circle. The other follows a rhumb line. Discounting adverse weather, the great circle route will win, every time.
I second the plane idea. Though actually the seat doesn't matter since he'll want to look out both windows to see the drop in both ways. Else he can just say, "The plane was tilted."
The connection between height and distance that this shows is why old ships used to have a crows nest. I love that connection, and you know how to do the calculation for why it matters. But proving that it is right is harder, particularly if you don't live near a large body of water.
With a bit more trig and a map you should be able to figure out the angle above the horizontal that a mountain somewhere in the visible distance should be. That angle should not be the same as it would be if it was flat. Verify it. Go driving somewhere else, verify it again using the same mountain.
But honestly, I'm personally most convinced by the fact that a phone call to someone in a different time zone makes it easy to verify that you're pointing different directions relative to the Sun.
As much as people like to point to things like eclipses, that relies on knowledge that you cannot verify yourself. I prefer verifications that you can duplicate with direct observation, without relying on outside experts. Because flat Earthers do not trust experts.
The Earth is roughly 6370 km which is not far from 6400, so if your eyes are 2m = 0.002 km up then the horizon is about sqrt(264000.002) = sqrt(25.8) km away, which is roughly 5 km or a bit over 3 miles.
If you apply this to a 6 km tall mountain, the horizon is about sqrt(263706) km away which is about 276.5 km. So something of the same height at the opposite end of the horizon would be 553 km away. So the top distance of 538 km is pretty close to the maximum that we would expect.
What about an airplane? An airplane flies about 11 km up. So it can see around 375 km. If you work it out, that puts the horizon about 3.37 degrees below horizontal. This isn't much, but if you take a plumb line and a right angle on an commercial flight, it is enough to actually see that the horizon is below horizontal.