Solution to "The Spiral Puzzle"

If we think about the spiral we are describing as proceeding from the center outwards, then every time it completes a revolution (2*pi radians), the radius of the spiral has increased by 1/4 inch. Therefore, the appropriate spiral formula to use, relating r, the distance from the center, to theta, the angle of revolution, is

r = 1/4 * theta / (2*pi).
Next, we realize that the number of revolutions we can make will be 48, since each revolution advances us 1/4 inch from the center, and we have to advance a total of 2 feet to reach the outer surface. Hence, the length of the spiral is
length = Integral ( 0 <= theta <= 48*2*pi) 1/4 * theta / ( 2*pi) d theta,
which gives us
length = 1/(8*pi) * ( 1/2 * (48*2*pi)**2 )
or
length = 2304 * pi inches = 192 * pi feet = 603 feet approximately.
Here, we are assuming that the length of the spiral is a good estimate of the length of the sheet. Of course, a spiral is linear like a string, while a sheet has thickness, and can be inherently curved, with the outer side being longer than the inner side.

Now if the manager doesn't believe all your high-falutin' math, you can simply lean back in your chair, secure in the knowledge that you are right, or you can repeat everything you said but LOUDER, or you can try a different way of attacking the problem that allows your audience to follow you.

A reasonable way of estimating the behavior of the problem is to suppose that, if we have a perfect cut, that all the wood in the log goes into the wooden sheet, and that the curvature of the wooden sheet can be ignored, so that it can be regarded, actually, as a very long, very thin box. In that case, to determine its length, we can ignore the cutting process, and pretend that the whole log was simply "melted" and then poured back out into the shape of a sheet.

Now we simply have to note that the volume of the log must equal the volume of the box, so, remembering that 1/4 inch = 1/48 foot, we have

( pi * 22 * 10 ) ft3 = length * 1/48 * 10 ft3
which implies that
length = 603 feet approximately.

Back to the The Spiral Puzzle.

Thanks to my brother Fred Burkardt for asking me and not believing me.

Last revised on 27 September 2000.