In wondering how his Welsh Corgi determined the fastest path to reach his ball, Tim Pennings of Indiana University used calculus. He assumed that the dog’s approach to this problem was to “minimize the time by minimizing the distance travelled” (Pennings 1), and as an equation, he used T(y)=z-yr+x2+y 2 s , where T(y) as the time it takes for his dog to reach the ball taking the path A to D to B, which Pennings assumes is the path his dog will take (1). The total distance from A to C is expressed as z, and r is the running time while s is the swimming time. This equation is in conjunction with Figure 1.
Figure 1: Paths to the Ball, Do Dogs Know Calculus?, Pennings, Page 178.
A is the point from which the ball is thrown at the edge of a lake, B is the final resting place of the ball in the lake, and all other points are parts of paths the dog can take to reach the ball, with the path from A to D to B skirting the lake and then crossing it starting at D, the path from A to C to B entering the lake at a right angle at C, and the path A to B swimming the entire distance to the ball.
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Pennings then took his dog to the park and threw the ball into the lake. He found that the dog took the path with point D, and using his dog’s three fastest swimming and three fastest running times, he found that dog’s fastest average running time was 6.40 meters/second, and his fastest average swimming time was 0.910 meters/second. Next, Pennings plugged these averages into the equation above and simplified to find y = 0.144x. This proportion would be true if Pennings’ dog took a mathematically perfect