THE SPIRAL PUZZLE
At a lumber mill, some logs are processed through a machine that essentially pares away the wood in a continuous spiral sheet, almost as though the log had been a roll of wrapping paper.
A certain amount of wood is wasted during the cutting process, especially if the sheet tears and the cutting process has to be restarted. Another source of waste occurs because the process must be terminated before the log is completely "unwrapped", leaving a scraggly pole that is tossed into the scrap bin.
An efficiency-minded manager believes that the amount of wood wasted in this process is as much as 25 percent. He is interested in confirming his guess by having you give a reasonable estimate of what the process would look like if it worked perfectly.
So suppose we begin with a log that is a perfect cylinder, 10 feet long, and 4 feet thick, and that we wish to spiral cut it into a continuous sheet that is 1/4 inch thick. We may assume that the sheet can be flattened, although it actually has some curvature that increases dramatically as the cut approaches the center of the log. Let us also assume that, in the messy rough and tumble of a lumber mill, it is typical to get several 1/4 inch thick sheets of wood that are equivalent to one sheet that is about 120 feet in length from this log.
The first question is, can you estimate or compute exactly the length of the sheet of wood you could cut from the log in a perfect mathematical process? It would be satisfactory to compute the length of a spiral line that represents the position of the cutting blade.
Secondly, believe it or not, the more formulas you write, the less some people believe you. Can you think of a simple way of getting a good estimate of the length of the sheet of wood that involves nothing more than arithmetic and geometry?