Do Dogs Know Calculus With Early Transcendentals?

Summary

A standard applications problem in differential calculus involves minimizing the travel time needed for a dog on a beach to reach a toy floating in the water, taking into account the different velocities the dog can achieve when running versus swimming. The solution usually presented to such problems are formulated in terms of a Cartesian coordinate. Here, a solution that is based in trigonometry is presented that, while quite natural, appears to be relatively unknown. The solution to the optimization problem in the trigonometric setting has a clear interpretation which is lacking in the Cartesian solution.

Acknowledgment

The author wishes to thank Rob Beezer at the University of Puget Sound for his suggestion that this solution should be better publicized.