## 73. Man and Dog.

- Walking the dog, - a mathematician friend once told me, - gives me plenty of food for thought. Once, for example, my dog, after waiting for me to go out, looked to see which way I was going to head, and when I started down the path, he raced to the end of it. Then he returned to me, ran to the end of the path again, and returned, and so on, doing this 4 times in total. All this time he moved at a constant speed, and when he ran to me for the last time, he covered the remaining distance of 81 m. Measuring the distance from my door to the end of the path later, I found it to be 625 m. At what speed did my dog run, if I was walking at a speed of 4 km/h?
73. The dog ran at a speed of 16 km/h. The key to solving the problem lies in the following considerations. The distance the man had left to walk alongside the dog was 81 m, or \(3^{4}\) (the dog returned 4 times), and the length of the path was \(625 \mathrm{m}\), or \(5^{4}\). Therefore, the difference in speeds (expressed in km/h) between the man and the dog (i.e., 12) and the sum of their speeds (20) should be in the ratio \(3: 5\).