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Nice View (Posted on 2004-09-29) Difficulty: 4 of 5
Assuming that the earth is a perfect sphere, in units of the earth's radius, how high must one be to see exactly one half of the earth's surface?

Okay... okay... how about exactly one third of the earth's surface?

No Solution Yet Submitted by SilverKnight    
Rating: 3.1667 (6 votes)

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Solution Solution + generalized | Comment 2 of 7 |

Of course the answer to seeing exactly on half of the earth’s surface is that you can’t. You’d have to be at infinity miles away because your two line of sight lines would be at the equator, so they would be parallel and never meet.

Basically what we are dealing with is this: if you pick a point P in space, the "border" of the portion of earth that you can see is created by drawing many lines that pass through P and are tangent to the surface of the sphere. In other words, the border is a circle. So we are dealing with the surface area of a spherical cap. The formulas for the area of a spherical cap and a sphere are:

C = 2 Pi rh = 2 Pi r(r-x) = 2 Pi r^2 – 2 Pi rx

S = 4 Pi r^2

(where h is the height of the cap, and x is the distance from the center of the sphere to the cap)

Well, in order to find at what x the cap is 1/3 the area of the sphere, we can solve for C = S/3.

2 Pi r^2 – 2 Pi rx = 4 Pi r^2/3
2rx = (2 – 4/3) r^2
2x = 2/3 r
x = 1/3 r

Alrighty, now call the center of the flat side of the cap A. Remember that the line from point P is tangent to the edge of the cap. Call this point T. And I’ll call the center of the sphere O.

So AO = x

OT = r

And we’ll call OP = H

Notice that AOT is a right triangle, and so is triangle TOP. Calling the angle between AO and OT alpha, we see from triangle AOT that cos(alpha) = x/r. From triangle TOP we can also see that cos(alpha) = r/H. So x/r = r/H. H = r^2/x = 3r.

So the height off the surface of earth is H – r = 2r. So that’s how high up you need to be to see 1/3 of the earth’s surface: 2r.

Generally speaking, if we want to know what the height is to see a portion, 1/f, of the earth’s surface, where 2<f, the solution is:

2 Pi r^2 – 2 Pi rx = 4 Pi r^2/f
2rx = (2 – 4/f) r^2
x = (1 – 2/f) r
x = r (f – 2)/f

x/r = r/H will still be true, so H = r^2/x = fr/(f-2)

And the real answer is H – r = fr/(f-2) – r = r[f/(f-2) – 1] = r[f-(f-2)]/(f-2) = 2r/(f-2)

  Posted by nikki on 2004-09-29 13:50:08
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