[Avodah] trig problem

[Micha Berger]

On Mon, Sep 24, 2012 at 11:06:02AM -0700, saul newman wrote:
: ...                                                i  knew  there were
: poskim like the CI talking about lateral movement  3 tfachim.   i cant
:  imagine a wind  taking it  vertically up 3  tfachim, but i could imagine a
: suspended from the top  sheet being lifted 3 tfachim up  when swayed a
: certain number of degrees from the vertical  ie  the hypotenuse of this
: triangle would  be missing  more than 3 tfachim when measured vertically
: from where it has been angled.  i wonder if someone could do the math ,

You seem to assume the top is tied to the pole, and it's the bottom
that isn't stable.

Let's start with the first extreme: If the billowing is on top, then
the fabric is going straight out, and then down. And so for the facric
to rise 3 tefachim, it would be billowing out 3 tefachim.

The hypoteneuse is always shorter than the sum of the legs of a right
triangle. After all, it's the "straight line" which is "the shortest
distance between two points", whereas going down and then across to
reach the same point is not. Unless the billowing is straight out from
where it's tied to the top pole.


To deal with the other cases, I need to convert to a single unit
of measure. So, I will speak of a 96" high sukkah, where the fabric
is raised 9" (appx 3 tefachim a/k/a 1/2 ammah).

The other extreme is that the billowing is on the bottom. You have an
96" hypoteneuse -- the fabric, and the top majority of the wall that
has fabric makes for a 96-9 = 87" vertical side. So your question is
what would be the length of the horizontal side.

87^2 + x^2 = 96^2 -- Pythagorean theorem
x = 40.6" = 3' 4.6"

That's billowing out pretty far!


The middle case would be if the billowing were halfway up. See this
"picture", in a fixed-width font
   |\
   | \
   |-->
   | /
   |/

The vertical line is where the wall should be, the diagonals trace
the fabric, and the horizontal line is the widest point of the bulge.
We can just divide the problem. Each triangle accounts for half the
height, so the vertical side for one triangle is (96-9)/2 = 43.5".
Each triangle also takes up half the fabric, so your hypoteneuse is
96 / 2 = 48" (4').

So, it's
43.5^2 + x^2 = 48^2
x = 20.3" = 2' 2.3"

IOW, the lower the billow the further out it has to be to raise the
fabric.

And that's being silly, and assuming the billow is more like a
triangle than some weird curve.

So, if it billows out 3 tefachim, it won't take up 3 tefachim of the
height of the fabric.

Of course, fabric doesn't really move in triangles, and in fact
ripples chaotically in the wind. So all this math is only to give a
ballpark feel of an answer.

GCT!
-Micha

-- 
Micha Berger             You are where your thoughts are.
micha at aishdas.org                - Ramban, Igeres Hakodesh, Ch. 5
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