For The Mathematically Inclined

If you drew a dot on the edge of a wheel and traced the path of the dot as the wheel rolled one complete revolution along a line, then the path formed would be called a cycloid , combining both forward and circular motion. What is the length of the path formed by one complete revolution? Assume the wheel has a radius of 1.

A town consists of only one street in the form of a circle. The town authorities give out four licenses for a particular kind of business. The inhabitants of the town live in equal density along the circle and will always go to the closest business for what they need. Business A gets to choose a location first, then business B, then C, and finally D. Each business desires to carve out as much business for themselves as possible but each knows the others all have the same motive. Assume that if a business is indifferent between locating in two different sections of the circle it will choose a section at random. Also assume that the business that goes last will choose a location in the middle of the largest (or one of the largest) sections. Where should business B choose relative to the location of A?

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PROBLEM #1
Find the coordinates of the point where the line through (3, - 4, - 5) and (2, - 3, 1) crosses the plane, passing through the points (2, 2, 1), (3, 0, 1) and (4,-1,0)

PROBLEM #2
What is the area of the triangle ABC with vertices A (1, 2, 3), B (2, -1, 4) and C (4, 5, -1)?