The solutions for each section are in a separate PDF file.
Arc Length Contest
Almost everyone either used the quadratic 6x-6x^2 or else the isosceles triangle with midpoint 1/2 and height 2. The lengths of these are roughly 3.249 and 4.123 respectively. Note that 6x-6x^2 is most likely the equation of the first diagram in the textbook. Rarer entries were (pi/2)sin(pi x), submitted by a couple people (length about 3.366) and 12(x^2-x^3), submitted by one person (length about 3.778). Only one person, however found the equation of the the second diagram in the textbook, which has length about 2.919. This is a half ellipse with equation (8/pi)Sqrt[x-x^2]. This student was Christopher From, who thus wins the contest. Congratulations!
Believe it or not this is not the best that can be done. See this article for an excellent discussion of the arc length contest. I hope you enjoyed it!