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Question
If a hypothetical building were to be built in the city of London, UK, how tall would it have be to be seen from Coventry, West Midlands UK, some 92 miles distant. You can use concepts of Pythagoras obviously but there is 'looking over the horizon' to consider also.
Many Thanks.

Answer
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The circumference of the earth is 24901.55 miles.

This means the angle between London and Coventry is the same fraction out of 360 as 92 is out of 24901.55. In other words, the angle from London to the center of the earth to Coventry is about 1.33 degrees.

This means the distance (straight line, burrowing through the earth, line A in the diagram) from London to Coventry is given by the law of cosines to be:

A^2 = R^2 + R^2 - 2 R R cos(1.33 degrees)

where R is the radius of the Earth (3959 miles). This can be simplified and computed:

A = R √[ 2 ( 1-cos(1.33 degrees) ) ] = 91.9

Notice that it's not much different than the distance by land, but it is a bit shorter.

Now, the angle that line A makes with the blue lines (which lead to the center of the Earth) must be (180-1.33)/2 = 89.335, and since at Coventry one must look at a right angle with this blue line, this smaller angle between lines A and B is 90-89.335 = 0.665 degrees.

(Note that this diagram is clearly not to scale, it would be too difficult to distinguish line A from line B or the arc of the Earth here.)

Likewise, the angle between A and C is 180-89.335 = 90.665, and so the angle between C and B must be 180 - 90.665 - 0.665 = 88.670.

We have all the angles and one side of the triangle ABC, that's enough to know all its dimensions. In particular, to get A, we can use the law of sines to get:

A/C = sin(88.670)/sin(0.665)

and so:

C = A sin(0.665)/sin(88.670) = 1.067 miles = 5633.8 feet


By all accounts, London is too far from Coventry to see (at least, while standing on the ground in Coventry -- things change if you get a "boost" to see higher). Typically a big city's skyline (which contains buildings roughly 1000-2000ft high) is visible for 40-70 miles, further if the geography or circumstances put the viewer higher than ground level and not as far if there is fog or if the city is higher up (i.e. on a hill).

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Clyde Oliver

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From "Advanced Math" -- I can answer all questions up to, and including, graduate level mathematics. I am more likely to prefer questions beyond the level of calculus. I can answer any questions, from basic elementary number theory like how to prove the first three digits of powers of 2 repeat (they do, with period 100, starting at 8), all the way to advanced mathematics like proving Egorov's theorem or finding phase transitions in random networks. I have no idea why Topology is not included in "Advanced Math" -- not to mention, there is only one question in this category, and it is decidedly not a topology question.

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