Re: I'd never seen this before

Tina <tbaker27705@xxxxxxxxx> wrote:
The equation in error was written as

sqrt (R^2 + d^2) = R + h

What error? That is the Pythagorean theorem.

Get a piece of paper, a straight edge, something to draw a big circle
and draw it out.

The line that represents the tower is of total length R + h, i.e. the
radius of the circle plus the tower height and is the hypotenuse of a
right triangle.

The other two sides of the triangle are:

The distance to the horizon, d, which is the line tangent to the circle
and whose length is from the tangent point to the top of the "tower".

The radius line, R, that intercepts the point where the distance line
intercepts the circle whose length is the radius of the circle.

I demonstrated this is an error by setting R and d to 1. The square
root of the sum of those squares is 1.414, yet the equal sign would
indicate it shoud be 2.

That is straightforward.

How is that?

Substituting 1 for R and d in the equation gives:
sqrt (1^2 + 1^2) = 1 + h
Evaluating 1^2 gives:
sqrt (1 + 1) = 1 + h
Adding 1 + 1 gives:
sqrt (2) = 1 + h
Taking the square root of 2 gives:
1.41 = 1 + h
Subtracting 1 from both sides gives:
..41 = h
QED

Never the less, the final answer that was posted was correct. As it
happens, that identity was not used later when the correct answer was
derived.

What identity? The whole thing is the Pythagorean theorem.

But both that post, and mine, provided the correct final equation, for
the geometric line of sight, but as to the visual one, it's subject to
both the assumptions that were llisted, and some that were not.

Yes.

The good news is, ain't no pilot gonna worry about this stuff. No
airplane pilot, anyhow,but those who pilot boats on blue water think
about those things pretty often, We use them, for example, to
estimate distances off shore.

Yep, and there are any number of rules of thumb to get a ballpark estimate.

--
Jim Pennino

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