From - Sat Jul 25 18:55:24 1998
From: ilan@isgtec.com
Newsgroups: rec.puzzles
Subject: Re: French Toast puzzle
Date: Wed, 15 Jul 1998 02:46:20 GMT
Message-ID: <6oh55s$r16$1@nnrp1.dejanews.com>
Geoff Hazel wrote:
> I'm cooking French Toast with uniform, square bread slices. I notice that
> I can just fit three pieces in a round skillet, with corners touching the
> edges, but no bent or cut or overlapped pieces. They form a "t" in the
> pan.
>
> I wonder with a pan 10 inches in diameter, what is the size of the bread?
>
> PS I don't have the answer, just the problem. good luck!
>
SPOILER
+---------+---------+
|\ x | |
| \ | |
| \ |y |
| R \ | |
| \|O |
| /| |
+----+---/+----+----+
| / |
| / |
| / R |
| / |
|/ |
|/ |
+---------+
Let O denote the centre of the circle, and R be its radius. Let x be the side
of each square, and y be the distance from the top of the "T" to the centre of
the circle.
Using the pythagoras theorem we have:
x^2+y^2 = R^2
(x/2)^2+(2x-y)^2 = R^2
Using the first equation, y can be eliminated from the second equation:
(x/2)^2+(2x-sqrt(R^2-x^2))^2 = R^2
Expanding and gathering terms yields
(13/4)x^2-4x*sqrt(R^2-x^2) = 0
This can be simplified into
(13/16)x = sqrt(R^2-x^2)
Raising to the second power and gathering terms finally yields
x = (16/(5*sqrt(17)))R
For R = 5" we get x = 16/sqrt(17)" = 3.88057".
__/\__
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__/\\ //\__ Ilan Mayer
\ /
/__ __\ Toronto, Canada
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