Why do this problem?
This problem offers an authentic context within which to calculate arc lengths and requires students to present their findings in a convincing manner.
This printable worksheet may be useful: Track Design
Arrange the class in groups of three or four, and hand out this worksheet to each group.
"Your task is to work out the dimensions of a running track that satisfy all the criteria on the worksheet, and to work out where to mark the start lines for the 200m (and/or 400m) race."
Give the students plenty of time to discuss and work on the problem in their groups. For groups not used to working collaboratively, these roles may be useful to guide students in organising themselves to work together.
To finish off, results could be presented as a list of the dimensions/angles that a groundskeeper would need in order to paint the lines for the running track, together with explanations of how students worked them out.
Additionally, students could be asked to produce a scale drawing of the track design to an agreed scale (1cm to 2.5m would fit on a large flipchart sheet) and then each group's drawing could be overlaid on another group's to check to see if they coincide.
If a straight line section is 85m, what is the length of a curved section, and what is its radius?
If you moved from the first lane to the second lane, how much further would you run on a complete lap of the track? What about in the third lane? Or fourth lane...?
By working in groups, students are encouraged to support each other in making sense of the problem and working towards a solution.
In reality, the 400m is measured 30cm in from the inside edge of the inside lane and 20cm in from the inside edges of the other lanes. Students could be given this extra layer of complexity to include in their working.
Some runners might complain that the sharper bend on the inner lane makes it more difficult to run around. Can students design a track which would reduce these effects?