2. Vasya and Petya, participating in a school sports and entertainment game, need to cover a distance of 3 km as quickly as possible with only one pair of roller skates between them. They start simultaneously, one just running, the other running on roller skates. At any time, the one running on roller skates can leave them to their friend and continue running without them. Such an exchange can be made as many times as needed. Find the minimum time to complete the distance (determined by the last one to arrive), if Vasya's speeds for simple running and running on roller skates are 4 km/h and 8 km/h, and Petya's speeds are 5 km/h and 10 km/h. Assume that no time is lost when switching between roller skates and running. (12 points)
Solution: The entire distance is divided into several stages, where one of the schoolchildren runs without roller skates, and the other on roller skates. Let $x$ be the sum of the lengths of the stages that Vasya runs on roller skates, $t_{1}$ be the time he spends running the entire distance, and $t_{2}$ be the time that Petya spends running the entire distance, then $t_{1}=\frac{x}{8}+\frac{3-x}{4}=\frac{6-x}{8}$ and $t_{2}=\frac{3-x}{10}+\frac{x}{5}=\frac{3+x}{10}$. Note that the function $t_{1}(x)$ is decreasing, while the function $t_{2}(x)$ is increasing. Let's specify the value of $x$ at which $t_{1}=t_{2}$, that is, the runners would finish the run simultaneously. From $\frac{6-x}{8}=\frac{3+x}{10} \Rightarrow x=2, t_{1}=t_{2}=\frac{1}{2}$ hour. If $x\frac{1}{2}>t_{2}(x)$, if $x>2$, then $t_{1}(x)<\frac{1}{2}<t_{2}(x)$. Therefore, the minimum time for the friends to complete the distance (by the last one to arrive) is 0.5 hours. Answer: 0.5 hours.