[TAG] Ya know, I've changed my mind.

[Benjamin A. Okopnik]

On Sat, Nov 05, 2005 at 07:57:30PM +0100, Karl-Heinz Herrmann wrote:
> Hi,
> 
> On Sat, 5 Nov 2005 10:03:44 -0700
> "Benjamin A. Okopnik" <ben at linuxgazette.net> wrote:
> 
> > > > 15.4 = sqrt( height^2 + ( height * ( 16 / 9 ) )^2 )
> 
> Well I didn't start with the sqrt form (buts it not terribly complicated
> then either ,-), but maby thats just a physicist opinion. 
 
Right - what do physicists know about math, anyway? :)))
 
> lets put d,h,w for diagonal, height and width.
> Then the great  Pytagoras said long ago (and somebody wrote it down
> obviously):
> 
>   d^2 = h^2 + w^2                  (1)
>
> so -- now you have your aspect ratio of 16:9 and once we know h, we
> also know w: 
> 
>  w=16/9*h                          (2)
> 
> substituting (2) in (1):
> 
>  d^2 = h^2 + ( 16/9*h )^2          (3)
> 
> or when we calculate that last term using 
> 
>  (a*b)^2=a*b*a*b=a^2*b^2           (4)
> 
>  d^2 = h^2 + (16/9)^2 * h^2        (5)

Ahhh! *That's* where you factored it! That's the part that I was
missing; I know the classic equation and can solve it for any of the
three factors when I have two of them (or even use it to make life
easier when multiplying numbers in my head - it cuts a step out of
multiplication), but got lost when I only had a _relationship_ between a
and b. My 5th-grade teacher would have been ashamed of me... but that
was long ago and far away. :)

> In (5) we have a sum of produkts -- and both summands have the common
> faktor h^2 so we can (German phrase is Ausklammern) "pull it out":
> 
>  d^2 = h^2 * ( h^2/h^2 + (16/9)^2 * h^2/h^2)

Yep. The rest is just a matter of reducing it.

>  d^2 = h^2 * (1 + (16/9)^2 )
> 
>  now divide by (1+..)  and get:
> 
>  d^2 / (1+ (16/9)^2) = h^2
> 
> take the sqrt on both sides:
> 
>  h = sqrt( d^2 / (1+(16/9)^2) )
> 
> with d=15.4 in we get:
> 
> perl -e 'print sqrt((15.4**2/(1+(16/9)**2)))."\n"'
> 7.55002309034141
> 
> perl -e 'print eval(sqrt((15.4**2/(1+(16/9)**2)))*16/9)." \n"'
> 13.4222632717181
> 
> which are in inch and the same numbers Bens iterative script got. 
> 
> For the paranoid:
> perl -e 'print sqrt(7.55**2+13.422**2)." \n"'
> 15.3997592188969
> 
> there are our 15.4 in diagonal again.
> 
> 
> 
> Now -- if you look at Bens original equation it says:
> 
>  d = sqrt ( h^2 + w^2 )
> 
> which is simply (1) with sqrt taken on both sides, so not actually
> evaluated on the right hand side.
> 
> 
> And if things get any more complicated can we please switch to TeX code?

[chuckle] I, for one, would have to spend a long while learning it; if I
was going to do that, I'd spend the time refreshing my algebra. Thanks,
Karl-Heinz!


* Ben Okopnik * Editor-in-Chief, Linux Gazette * http://linuxgazette.net *