Quote:
For height to avoid earth curvature, one of the math types here should know. I think 30' should be sufficient, though.
Ooh. That's a good one.
First, we need to determine the angle of the vertex at the center of the earth to one of the stations and the midpoint (that is, high spot) of the line connecting the two stations. We know the length of all three sides, so we can use the oblique triangle formula: a^2 = b^2 + c^2 - 2bc(cosA). b and c are the radius of the earth (call it r for easier calculations, and assume the maximum radius -- 6379km -- for head room) and a is half of the 22 miles (call it 18km for a little head room, shown as "d" in the picture). Give that b=c=r, the formula reduces to a^2 = 2r^2(1-cosA) and solving for cosA leaves cosA = 1 - (a^2/2r^2). Add in real numbers and you get cosA = 0.999996019... and then A = 0.1616749292 degrees. (Actually, because of the next step, we don't even need to resolve the actual angle.)
Now we need to find the length of the hypotenuse of a right triangle with one angle being A from above and the side adjacent to A being the radius of the earth. We know that cosA = adjacent/hypotenuse. So cosA = r/h. We know A and r, so solve for h: h = r/cosA. Plug in real numbers and you get h = 6379.025396, or 0.025396km longer than the radius of the earth. That's over 25 meters, or over 83 feet. Significant. That's 83 feet on each side. I hope you're living in a tall apartment building.
Edited by wfaulk (01/08/2007 13:25)
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Bitt Faulk
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