the beach race

A beach race begins from a point, 4 km out to sea from one end of a 10km beach. Racers must swim to a point along the beach, and then run to the finish line. If I can swim at 4 kph and run at 10 kph, to what point on the beach should I aim to land?

Can you find the landing point for which the time of the race will be as short as possible - as accurately as possible?

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

   
   





2. There are two landing points which will result in a race time of exactly 2 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 hour race.

   
   




3. Now build an algebraic model for the times for the race, letting the distance OC = x:
   
     swimt(x) =
   
     runt(x) =
   
     racetime(x) =




4. Finally, plot the derivative of this RaceTime(x) function to locate the best place to land:
     Derivative[RaceTime(x)] =  

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

This problem may also be modelled using CabriJr on the TI-83/84 series calculator. Click on the graphic below to download the file and try it yourself!

Try having students work in pairs: one generating data points using the CabriJr model and the partner entering these into lists to plot and then to help build their algebraic model!

And just wait until you try this with TI-nspire CAS!