The Spiral Puzzle
Solution

Solution 1: 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.

This means we can figure out a formula that relates r, the radial distance from the center, to theta, the angle of revolution:

        r = 1/4 * theta / (2*pi).
      

Using the polar coordinates that are natural to this problem, we need to estimate the infinitesimal element of arc-length along the spiral. Although we're actually on a spiral, we'll approximate by assuming that we're traveling on a circle, in which case, the element of arc-length is

        ds = r dtheta
      

Therefore the total length of the arc is

        S = Integral ds 
          = Integral r dtheta 
          = Integral 1/4 * theta / (2*pi) dtheta.
      

To establish the integration limits in terms of our chosen integration variable theta, we realize that the number of revolutions we make will be 96. This is because, starting at r = 0, each revolution advances us 1/4 inch from the center, and we have to advance to r = 2 feet, or 24 inches, requiring 96 turns.

Hence, our integral may be written as

        S = Integral ( 0 <= theta <= 96*2*pi) 1/4 * theta / (2*pi) d theta
      

This gives us

        S = 1/(8*pi) * ( 1/2 * (96*2*pi)**2 ) inches
          = ( 192 * 12 * 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.

Solution 2: 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

length = 603 feet approximately.

Back to The Spiral Puzzle.

Thanks:
to my brother Fred Burkardt for asking me and not believing me, and
to James Bainter, who pointed out a pair of arithmetic errors.