Originally Posted by Tony Szamboti
As I already stated, the tilt does not produce the misalignment necessary for the columns to miss each other by its rotation either. A 1 degree tilt produces less than an inch of horizontal shift over 207 feet.
The horizontal shift would be the difference between the adjacent side and the hypotenuse of a 1 degree angle in a right triangle.
The towers were 207 feet on a side above the ninth floor and 207 feet is equal to 2,484 inches.
Take the cosine of 1 degree and multiply it by 2,484 inches to get the length of the adjacent side. Then subtract that from the hypotenuse which would be 2,484 inches and you will have your answer.
OK, that's where I started, although I was doing a different function and using the tangent. (not to mention I had a calculator that used radians, not degrees, which was part of my problem) Technically, it's not a right triangle, because if the thing tilts, you'd have to calculate an arc, but that's beyond me. Right triangle is close enough for jazz.
OK, so the cosine of 1 degree is .0.9998476951563913, and multiplying that times the hypotenuse I get 2483.6216747684757" for the length of the second side.
Using the Pythagorean Theorem, I get the third side to be ~43". Checking the math, this bears out. Again, please correct me if I'm wrong.
A^2 + b^2 = c^2
Small side^2 + adjacent side^2 = hypoteneuse^2
43.351776^2 + 2483.6216^2 = [2,484^2] OR 6,170,256
1879.376482354176 + 6168376.623379768 = 6170255.999862122 ~6171256
Square root of the number on the right gives me ~2,484 (2483.9999)
If I try this with your number of "less than an inch," I can't make it work. As far as I can tell, for a 1 / 89 / 90 triangle, with the short side being 1", side B would be 57.289963" and the hypotenuse would be 57.298687".
Again, please show the errors in my math above. I just can't validate your statement using my limited knowledge of trig.
And, again, the towers were built in 3 dimensions, so the swing could have been north/south or east/west or, more likely both over that entire span.
Thanks.
I welcome corrections from all comers; it's been a while since I have done this stuff.
ETA - I just noticed where you said "subtract that [the adjacent side] from the hypotenuse" to get the third side. Are you using some different rule of thumb that I don't know?