Stephen ARNOLD
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the beach race

Learning Objectives

  1. Interpret the variables involved in using some simple non-linear functions to model situations and make related predictions.
  2. Use known formulas to develop, apply and interpret new relationships.
  3. [Calculus Extension, if appropriate] Apply the differential calculus to find the minimum value of a function.
A beach race begins from a point, 6 km out to sea from one end of a 9 km beach. Racers must swim to a point along the beach, and then run to the finish line. If I can swim at 3 km/h and run at 12 km/h, to what point on the beach should I aim to land?

In this task, you will attempt to find the landing point for which the time of the race will be as short as possible.

Sorry, the GeoGebra Applet could not be started. Please make sure that Java 1.4.2 (or later) is installed and activated. (click here to install Java now)


  • 1. Drag point C to find the landing position which will result in the LONGEST time for the race. [LO1]


  • 2. There are two landing points which will result in a race time of approximately 2.75 hours. One of these is to swim directly to the point O and then run the full length of the beach (swimming the shortest possible distance). Drag the point C to find the other landing point which will result in a 2.75 hour race. [LO1]


  • 3. I can swim at 3 km/h. In my training, I swam 12 kilometres. How long did that take me? [LO2]

  • 4. If we let OC = x kilometres, then find an expression for the distance BC (my swim distance) in terms of x. [LO1]

    swimdistance(x) =  

  • 5. Now find an expression for the distance AC (my run distance) in terms of x. [LO1]

    rundistance(x) =  

  • 6. Use these expressions to build an algebraic model for the times for the race, in terms of the distance OC = x: [LO2]

    swimtime(x) =

      runtime(x) =

      racetime(x) =



  • 7. Finally, plot the derivative of this RaceTime(x) function to locate the best place to land: [LO3]

    Derivative of RaceTime(x) =  



  • 8. The best place to land occurs when x =
     



  • 9. [EXTENSION 1]: The slider at the bottom of the applet window allows control of the swimspeed and runspeed - or, rather, the ratio of swimspeed to runspeed. Suppose we describe runspeed = k*swimspeed, what restrictions might realistically be placed upon the value of k for this problem situation?

  • 10. [EXTENSION 2]: For what value of k would my ideal landing place be exactly halfway along the beach (i.e. OC = AC = 4.5 km)?


 

Steve Arnold, with thanks to Mike May S.J. for his JavaScripting. 24/03/2008, Created with GeoGebra



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