Didn't bother. However, since all the outside surfaces are still there, and each cut has produced *two* inside surfaces, assuming an ideal knife and a perfect central cut the math is simple, doesn't require calculus, and will produce a result that's close enough for my caring.
Which is to say, the original is ROUGHLY the surface of a cylinder of length equal to the distance from any point in the center of the bread through the centre line of the bagel back to the origin - or, pi*d*H, where "d" is the (uniform) thickness of the bagel at the widest point (and H itself is 2*pi*R, where R is the distance from the centre of the bagel's hole to the middle point of the bread.)
The new exposed surface is ROUGHLY 2*d*H.
Therefore, old surface = pi*d*H New surface = 2*d*H. Total surface of new cut = (pi+2)d*H
Ratio of new surface to original surface = (pi+2)/pi
Or, roughly, 5/3.
I'm cheating and making my math simpler by assuming the changes induced by the curves of the cut are non-relevant compared to the cut itself. And I get to do that, because I'm an engineer. Precise answers are for people who aren't about to eat the subject of the calculation regardless of the accuracy of the result.
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(no subject)
Date: 2009-12-08 05:25 pm (UTC)(no subject)
Date: 2009-12-08 06:07 pm (UTC)Which is to say, the original is ROUGHLY the surface of a cylinder of length equal to the distance from any point in the center of the bread through the centre line of the bagel back to the origin - or,
pi*d*H, where "d" is the (uniform) thickness of the bagel at the widest point (and H itself is 2*pi*R, where R is the distance from the centre of the bagel's hole to the middle point of the bread.)
The new exposed surface is ROUGHLY 2*d*H.
Therefore, old surface = pi*d*H
New surface = 2*d*H.
Total surface of new cut = (pi+2)d*H
Ratio of new surface to original surface = (pi+2)/pi
Or, roughly, 5/3.
I'm cheating and making my math simpler by assuming the changes induced by the curves of the cut are non-relevant compared to the cut itself. And I get to do that, because I'm an engineer. Precise answers are for people who aren't about to eat the subject of the calculation regardless of the accuracy of the result.
(no subject)
Date: 2009-12-09 03:34 am (UTC)(no subject)
Date: 2009-12-09 03:34 am (UTC)(no subject)
Date: 2009-12-08 06:09 pm (UTC)(no subject)
Date: 2009-12-08 06:43 pm (UTC)(no subject)
Date: 2009-12-08 09:58 pm (UTC)(no subject)
Date: 2009-12-09 01:06 am (UTC)\m/-_-\m/
(no subject)
Date: 2009-12-13 04:11 pm (UTC)