Do dogs know calculus? A mathematician stands on a beach with his dog at point ?A. He throws a tennis ball so that it hits the water at point ?B.? The dog, wanting to get to the tennis ball as quickly as possible, runs along the straight beach line to point ?D and then swims from point ?D? to point ?B? to retrieve his ball. Assume point C is the closest point on the edge of the beach from the tennis ball (see figure). a. Assume the dog runs at m/s and swims at ?r? > ?s.? Also assume the lengths of BC, CD,? and ?AC? are ?x, y?. and ?z? respectively. Find a function ?T?(?y?)representing the total time it takes for the dog to get to the ball. b. Verify that the value of y that minimizes the time it takes to retrieve the ball is y = x . ? s+1? s?1 c. If the dog runs at 8 m/s and swims at 1 m/s. what ratio ?y?/?x? produces the fastest retrieving time? d. A dog named Elvis who runs at 6.4 m/s and swims at 0.910 m/s was found to use an average ratio ?y?/?x? of 0.144 to retrieve his ball Does Elvis appear to know calculus? (?Source: College Mathematics Journal,? May 2003)

Solution Step 1 Consider that a mathematician stands on a beach with his dog at point A . Consider that he throws a tennis ball so that it hits the water at the point B. Now, consider the movement of the dog. The dog starts running straight along the beach at point . The speed of the dog i m/s. Again from point D it starts swimming to the point . The swimming speed is s m/s. Also, at the edge f the beach, is the cl sest point t from the beach. Let us draw the scenario. Here angle Here the lengths are AC = z DC = y BC = x Step 2 (a)Consider the time function of the dog to get the ball isT(y) Now, con sider the running time. By running it passes A D. So, you can say AD = AC DC Put DC = yand AC = z AD = zy The running speed is r m/s. So, the time function for this is zy T 1y) = r Consider the swimming time. By swimming it passes B D. Now, according to Pythagoras theorem: PutDC = yand BC = x: The running speed is s m/s. So, the time function for this is y +x T 2y) = s So, the total time function is T(y) = T (1)+ T (y)2 zy y +x T(y) = r + s y +x So, the time function is T(y) = zy+ r s Step 3 (b)Now, verify that the value of y that minimizes the time it takes to retrieve the ball is y = r x r s1 s1 Consider the time function again zy y +x2 T(y) = r + s Find the derivative: Now, at minimum time: So, you get: