5. (10 points) The distance from home to work is $s=6 \kappa$ km. At the moment Ivan left work, his favorite dog ran out of the house and ran towards his owner. They met at a distance of one third of the entire path from work. The dog instantly turned around and ran back home. Upon reaching home, he instantly turned around again and ran towards his owner, and so on. Assuming Ivan and his dog move at constant speeds, determine the distance the dog will have run by the time Ivan gets home.
Answer: $12 \kappa n$

Solution. By the time of the first meeting, Ivan had walked one third of the entire distance. Therefore, the dog ran two thirds of the entire distance. That is, the dog's speed is twice Ivan's speed $v_{n}=2 v_{u}$.

(4 points)

On the way from work to home, Ivan walked $6 \kappa$.

Therefore, the dog ran $l=2 \cdot 6 \kappa n=12 \kappa n$.