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In sci.physics, Sam Wormley
<swormley1@mchsi.com>
wrote
on Sat, 19 Feb 2005 04:53:15 GMT
<_CzRd.9886$g44.8648@attbi_s54>:
[MATHTREK]
The Coin in the Cake
Slicing a cake to find a hidden coin is less chancy than you might think.
http://www.sciencenews.org/articles/20050219/mathtrek.asp
Especially if one uses a large mace, though I doubt that
is in any Greek (or other) tradition. :-) But with the
given values, the probability of hitting a coin whose
center is situated somewhere within the circle of radius
15 cm (otherwise the 1 cm radius coin would stick outside
the 16 cm radius cake) would simply be the area of the
"locus cross" or "star" with slightly rounded edges, divided
by the total area of the coin placement (the "cross" is
the locus of all center placements which touch the cuts,
which for 2 cuts are very conveniently placed on the
coordinate axes). The width of each arm is 2 cm (the
diameter of the coin), the length approximately 30 cm.
(The edge roundoff will lead to a small error, as will
the width of the cutting knife.)
For a 2-cut the probability isn't too hard to calculate, and
is approximately 116 / (Pi * 15^2) = .1641 .
Of course, I'm a computer engineer; therefore...
A simple Monte Carlo simulation using drand48() suggests
the probability is closer to .1543 . An even simpler
"step placer" suggests the same.
I'm a little puzzled as to where the paper got .2987. I
sense a minor r dtheta versus dtheta integration problem here.
For larger numbers of cuts the area of the "star" becomes
harder to calculate, as the center area has to be
accounted for properly. The probability becomes certainty
when the center area dominates the entire cake, or,
alternatively, when the slices become too thin to contain
the coin in its entirety even if the knife were to somehow miss.
I should note that a cake *is* a cylinder, in most cases,
though the problem statement assumes the coin is parallel to
the circular faces of the cake.
Perhaps I'm making an egregrious error here; if so, I'd
like to know what it is. But this problem is unfortunately
coming out, ahem, half-baked, at least for me.
(Disclaimer: I'm not Greek.)
This is more of a math problem than a physics one; followups
reset accordingly.
--
#191,
It's still legal to go .sigless.
.
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