## Task B-2.7.

Mirko is swimming from the northern to the southern, and Slavko from the eastern to the western shore of the lake. At the initial moment of observation, Mirko is 600 meters north, and Slavko 400 meters east of a buoy located somewhere in the middle of the lake. Both swimmers move uniformly in a straight line, Mirko at a speed of $1.6 \mathrm{~m} / \mathrm{s}$, and Slavko at a speed of $1.2 \mathrm{~m} / \mathrm{s}$. After how much time from the start of the observation will the distance between Mirko and Slavko be the smallest possible? Will both swimmers cross the level of the buoy at the moment when this distance is the smallest? Justify your answer.
## Solution.

![](https://cdn.mathpix.com/cropped/2024_05_30_7713364b1a15726862ddg-10.jpg?height=543&width=471&top_left_y=1813&top_left_x=724)

For $t$ seconds, Mirko swims $1.6t$ meters, and Slavko swims $1.2t$ meters.

Then Mirko is $|600-1.6t|$ meters away from the buoy, and Slavko is $|400-1.2t|$ meters away from the buoy.

The distance between them is $d(t)=\sqrt{(600-1.6t)^{2}+(400-1.2t)^{2}}$.

To determine at what moment the distance between Mirko and Slavko will be the smallest, we need to find $t$ for which the function $f(t)=(600-1.6t)^{2}+(400-1.2t)^{2}$ reaches its minimum. Note that this is a quadratic function $f(t)=4t^{2}-2880t+520000$, and we need to find the x-coordinate of the vertex,

$$
t_{0}=\frac{-b}{2a}=\frac{2880}{8}=360 \text{ s}
$$

After 360 seconds, or 6 minutes, the distance between Mirko and Slavko will be the smallest possible.

Mirko swims 1.6 $\cdot 360=576$ meters in 6 minutes, and Slavko swims $1.2 \cdot 360=432$ meters.

Since Mirko was initially 600 meters away from the buoy, and Slavko was 400 meters away, this means that after 6 minutes, when the distance between them is the smallest, Mirko has not yet reached the level of the buoy, while Slavko has already passed the level of the buoy.

## SCHOOL/CITY COMPETITION IN MATHEMATICS

3rd grade - high school - B variant
January 4, 2017.

## IF A STUDENT HAS A DIFFERENT APPROACH TO SOLVING THE TASK, THE COMMITTEE IS OBLIGED TO GRADE AND EVALUATE THAT APPROACH IN AN APPROPRIATE MANNER.