MATH 1121 Calculus

Project #4

Dogs and minimal times:

Several years ago, Tim Pennings made national news by examining (rather tongue-in-cheek), whether his dog must know Calculus.

Here's the article:

http://www.maa.org/features/elvisdog.pdf

I want you to read this article, and see that, in fact, you know everything you need to know to understand the whole thing.  Note the use he makes of the Second Derivative Test, and how he derives the general form of the solution.

Here's a Geogebra applet that allows you to play around with the distance along the shore, the position of the ball, the speeds the dog can run and swim, and the place where the dog decides to jump into the water:

Applet

Now, use the applet below to trace out the graph of T(J), the total time as a function of the jump-off point, for several different ratios, r/s. 

Applet

So here's your assignment:

1.  Using the values: r = 6.4, s = 0.9, z = 15 and x = 18, use calculus to actually find the place, J, that minimizes the total time, T, and find out what that minimum value of T is.

2.  One of the most valuable things in Dr. Pennings' paper is his list of simplifying assumptions that he makes in order to create the model.  See if you can think of a few more assumptions he had to make, in order to come up with his mathematical model.

3.  If you look at the scatter plot of his results, you notice that the fit is very good for smaller values of x, and not so good for larger values of x (the distance out from shore.)  By thinking about what the second GeoGebra applet shows, what about this fact might make you suspicious that the dog is not actually quite as good at minimizing the time as might at first appear?

4 (optional).  Dr. Pennings suggests that it is possible that the dog is actually better at optimizing the actual time required than the model might indicate.  What leads him to think this, and how could you test this hypothesis?

(Just so you know, I think this is one of the best real-life tests of a Calculus model that I've ever seen.  Dr. Pennings lays out the work and the thought process beautifully.)

Page Last Modified: 13 April, 2008

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