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Four Challenge Questions

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jmorgan@math.uh.edu

There are two rectangles on the right. The original one is depicted in blue. The yellow rectangle is the result of shrinking the blue rectangle in both the vertical and horizontal directions, rotating it, and repositioning it on top of the blue rectangle.

Question: Can you show that the rectangles have exactly one common point?

The solution requires trigonometry.

This problem was given to a large group of students who had never seen geometry. Many of them solved the problem (although not immediately!!).

The Problem: Divide the circle into at least three pieces so that all pieces are the same size and shape, and at least one of the pieces does not touch the center of the circle.

The solution requires thought.

The Problem: Give parametric equations governing the motion of the two wheels in the figure on the right.

Joe Smith tunes into the same radio programming for an average length of 30 minutes the same time each day, seven days each week. What he listens to is a pre-recorded program that loops continuously through the 7-day week (meaning it repeats over and over again.) How long would the total program have to last for Joe to hear a different 30 minute-portion each time he listens at the same time for a week? (Each week a new program is produced.)

Note: It is easy to see that 3.5 hours is the minimum amount of recording time. Suppose the station decides this just isn’t enough recording time and they want to know if there are other options. What are the other possible recording times which allow Joe to head a different show every day, while remaining under 24 hours of recording?