Auto: Rybnikov and..

A bus traveling a 100 km route is equipped with a computer that shows the predicted remaining time until arrival at the final destination. This time is calculated on the assumption that the average speed of the bus on the remaining part of the route will be the same as on the part already traveled. Forty minutes after the start of the journey, the expected travel time was 1 hour and remained the same for another five hours. Could this have been possible? If so, how many kilometers had the bus traveled by the end of these five hours?
Let $s$ km be the distance the bus has traveled $t$ hours after the start of the journey ($2 / 3 \leq t \leq 5^{2} / 3$). Then its average speed over the traveled part of the route is ${ }^{s / t}$. According to the problem, moving at this speed, the bus will travel $100-s$ km in one hour, that is, $s / t=100-s$. From this, we get that the bus should move according to the law $s=100 t / 1+t$. It is easy to verify that the function $s=100 t / 1+t$ is strictly increasing. Therefore, movement according to this law is possible. For $t$ $=52 / 3$ we get

$s=85$.

## Answer

It could; 85 km.