Board Problems

Answers do not suffice. We need full explanations.

  1. How would you go about estimating the diameter of the Earth? Can you replicate Eratosthenes?
  2. How would you go about constructing a table of sines? What mathematical facts do you need?
  3. Wrap a strip of paper around a candle. Then cut through the paper and the candle with a sharp knife (a power saw would be more impressive). What kind of  a curve is generated when you unroll the paper?
  4. When I flew to Helsinki a few weeks ago on the first leg of the MAA's Euler Tour, there was a display in our cabin of our altitude, external temperature, distance to our destination, local time at the destination, etc. Also shown was a map of the world with our current location. Part of it was shaded to show where it was night. What curve marked the boundary between day and night on this map?
  5. Find the time of high tide on the Potomac on each day of the year. Plot the times. Is your curve continuous? If not, why not? Can you fit a curve to these points? You do the same thing with the time of sunrise and sunset in DC.
  6. I don't know if you have ever encountered a mathematical crank or not, but it is an interesting experience (and can be a horrible waste of time). Here is an example. Or is it?