1. At the same time, a car departs from point $A$ to point $B$ at a speed of 80 km/h, and from point $B$ to point $C$ - a train at a speed of $50 \kappa м / ч$. After 7 hours of travel, they were at the shortest distance from each other. Find the distance between the points, if all three points are equidistant from each other and are connected by straight roads.
Solution: Let $AB = BC = AC = S$. Denote the distance between the car and the train as $r = r(t)$, where $t$ is the time from the start of the motion. Then, by the cosine theorem, we have: $r^{2} = (S - 80t)^{2} + (50t)^{2} - 50t(S - 80t)$. To find the time at which the distance between the car and the train was the smallest, we will compute the derivative of the function $r^{2} = r^{2}(t):$

![](https://cdn.mathpix.com/cropped/2024_05_06_9cd2bbfc846d490b2ee8g-08.jpg?height=428&width=476&top_left_y=503&top_left_x=1441)

$\left(r^{2}\right)^{\prime} = -2 \cdot 80(S - 80t) + 2 \cdot 2500t - 50(S - 80t) + 80 \cdot 50t = -210S + 2(6400 + 2500 + 4000)t = 0$.

Since the function $r^{2} = r^{2}(t)$ takes its minimum value during the motion at $t = 7$, then $-105S + (6400 + 2500 + 4000) \cdot 7 = 0$, and $S = 860$.

Answer: $860 \kappa m$.