Do Dogs Know Calculus ? This project will explore the innate ability of a dog to find the quickest path to retrieve a ball thrown into the water. We calculate an optimal path for a given situation based on given running and swimming speeds and compare with actual data from real trials. The concept, solution and data in this project are based on the work of Timothy Pennings of Hope College in Holland, Michigan.

Do Dogs Know Calculus ? B WATER A C D LAND Suppose Elvis the dog finds himself at point A in the diagram below and wants to retrieve a ball at point B, floating stationary in the water. Of course, the shortest distance from A to B is a straight line, but suppose Elvis can run much faster than he can swim. Should Elvis choose the path of shortest distance or, assuming Elvis wants to get to the ball as quickly as possible, is there some other path that will minimize the total time to reach the ball? B WATER A C D LAND

For example, suppose Elvis were to run along the shoreline to point D and then jump into the water and swim to point B. Is this alternative path faster? Is there a optimal choice for point D (between A and C) such that the total time taken from A to B is at a minimum? We can answer this question with basic calculus. Finding the answer is the purpose of this project. We can then compare the theoretical answer to choices made by Elvis, a real dog, owned by Tim Pennings, the originator of the concept for this project. B WATER A C D LAND

B A C D To answer with basic calculus, we’ll need to generalize the situation and simplify – for example, ignore the actual shoreline. Instead, assume the shoreline follows exactly the line containing segment AC. Assume that the ball stays put at B and assume that when Elvis jumps into the water at any point from A to C, he would immediately begin swimming. Assume that his running and swimming speeds will be constant.

B x A y C D z Lets call z the distance from A to C, y the distance from D to C, and x the distance from C to B. We’ll assume that for each toss of the ball that x and z are constant and x is measured perpendicularly to segment AC, the coastline. Essentially, we are interested in an optimal choice for y that is based on x and z. Lets assume that Elvis’ running speed is r and his swimming speed is s.

Question 1: Write an expression T(y) in terms of r, s, x, y and z that represents the total time to get from point A to D and then from D to B. Now that we have T(y) we seek a value of y that will minimize T. This will occur when T’(y) =0. Question 2: Find T’(y). Question 3: Solve the equation T’(y) = 0 for y. This gives us the optimal value for y for any value of x. Does the optimal value of y depend on z ? Question 4: Show that the solution in question 3 is in fact a minimum for T using the second derivative test. Question 5: Suppose the running and swimming speeds for Elvis are r = 6.4 m/s and s = 0.91 m/s, respectively. Plug these values into your solution from question 3 to find a linear function that gives optimal values of y as a function of x. We can call this solution the optimal line for r and s.

Question 6: We will now compare the theoretical optimal solution for Elvis with his actual choices in repeated trials. The table on the following page includes values for x which is the measured distance from the shoreline to the point B where the ball is floating. Next to the values of x are values of y which represent measurements of distances chosen by Elvis for each value of x. As before, y is the distance from D to C, as in the previous diagrams. The table contains 35 pieces of data, which according to Mr. Pennings, took 3 hours to complete. In spite of the length of the experiment, Mr. Pennings reported, throughout Elvis had no interest in stopping or slowing down. To answer question 6 sketch the optimal line found in question 5 and plot a few of the points from the table (creating a scatter plot.) Describe how well Elvis did in comparison to the optimal line, specifically with respect to larger and smaller values of x.

Results of 35 repeated trials: x = distance from C to B (the distance the ball is thrown) y = distance from D to C (corresponds to Elvis’ choice of optimal path) x y 10.5 2.0 17.0 2.1 4.7 0.9 10.9 2.2 15.3 2.3 7.2 1.0 15.6 3.9 11.6 11.2 1.3 11.8 10.3 1.8 6.6 11.5 15.0 3.8 7.5 1.4 11.7 1.5 14.0 2.6 9.2 1.7 14.5 1.9 12.2 13.4 13.5 6.0 12.7 19.2 4.2 6.5 14.2 0.8 11.4 2.4 2.5 12.5 3.3