2. (16 points) A truck left the village of Mirny at a speed of 40 km/h. At the same time, a car left the city of Tikhaya in the same direction as the truck. In the first hour of the journey, the car traveled 50 km, and in each subsequent hour, it traveled 5 km more than in the previous hour. How many hours will it take for the car to catch up with the truck if the distance between the village and the city is 135 km?
# Answer: 6.

Solution. In the first hour of travel, the pursuit speed of the car relative to the truck was $50-40=10$ km/h. In each subsequent hour, the pursuit speed increases by 5 km/h. Thus, the pursuit speeds form an arithmetic progression $10,15,20 \ldots$ km/h. Let's find the number of hours $n$ until the cars meet. The sum of this progression is equal to the total distance the car will travel until it meets the truck $\frac{20+5(n-1)}{2} \cdot n=135$. After transformation, we get the equation $n^{2}+3 n-54=0$, the roots of which are $6,-9$.