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Differentiation Area and Perimeter

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m...@privacy.net

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Mar 31, 2006, 3:30:01 AM3/31/06
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Maybe its too early in the morning and I haven't engaged the brain yet,
but I can't think about how to prove the following:

Given a rectangle with sides W and H
The perimeter P=2W+2H
The area A=WH

If A is fixed how do I show the minimum P occurs when W=H (ie a square)?

--
Timothy


m...@privacy.net

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Mar 31, 2006, 3:51:12 AM3/31/06
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In message <i0TGvmBJ...@town-village.demon.co.uk>, "m...@privacy.net"
<m...@Privacy.Net> writes
I can do it for a given P, I could then try proof by induction by
showing that it applies for P+1, then show it works for P=1.


--
Timothy


Charlie Turner

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Mar 31, 2006, 3:56:58 AM3/31/06
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But you want to show it for all P, not just integer P, don't you? I'm
working on it but feeling stupid myself.


m...@privacy.net

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Mar 31, 2006, 4:28:10 AM3/31/06
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In message <e0iqsn$j2v$1...@heffalump.dur.ac.uk>, Charlie Turner
<charlott...@durham.ac.uk> writes

I've got it:

A=WH
P=2W+2H
W=A/H

P=2(A/H)+2H

dP/dH=-2A/(H^2)+ 2

min at 0

0= -2A/(H^2)+2

H^2 = A

Therefore H=root A for min P
ie A square

How many marks is that worth in an essay about minimising building
costs?!

--
Timothy


Stu

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Mar 31, 2006, 6:00:35 AM3/31/06
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Substitute H=A/W into the first equation then solve the resulting
quadratic?

P=2W+2A/W -> PW=2W^2+2A -> 0=2W^2-PW+2A, then solve as appropriate.

Stu
--
From the prompt of Stu Teasdale


dae-...@at.dot.durge.org

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Mar 31, 2006, 8:37:56 AM3/31/06
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I know some very good answers have been given already, but here's how I'd
have done it...

Rather than considering the relationship between the full perimeter, work
from (x=)P/4; which would of course be the length of the side of a square
for the given area, W and L then become x +/- d:

A = W . H = (x-d)(x+d) = x^2 - d^2

Now, can anyone remember which value of d gives the lowest possible value of
d^2? :)

Cheers

- Darren

--
Darren Edmundson - Internet & 3G Technologist
MSN:grid...@hotmail.com


ne...@loowis.durge.org

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Mar 31, 2006, 9:08:45 AM3/31/06
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dae-...@at.dot.durge.org wrote in alt.dur.general:

> Now, can anyone remember which value of d gives the lowest possible value of
> d^2? :)

i
HTH

Andy Scheller

<can't be bothered with a sig>


Charlie Turner

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Mar 31, 2006, 9:23:33 AM3/31/06
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ne...@loowis.durge.org wrote:
> dae-...@at.dot.durge.org wrote in alt.dur.general:
>
>>Now, can anyone remember which value of d gives the lowest possible value of
>>d^2? :)
>
>
> i
> HTH
>
erm.... what about 2i, 5i, 905383476309103852764i?


Tom Joyce

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Mar 31, 2006, 1:06:45 PM3/31/06
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Also sprach Charlie Turner <charlott...@durham.ac.uk>:


Or even 54281905383476309103852764j?

--
yours aye,
Tom


Charlie Turner

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Mar 31, 2006, 1:35:58 PM3/31/06
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Tom Joyce wrote:

j? whassat? talk maths, not engineer!

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