4. At some point on the shore of a wide and turbulent river, 100 m from the bridge, Gavrila and Glafira set up a siren that emits sound signals at equal intervals. Glafira took another identical siren and positioned herself at the beginning of the bridge on the same shore. Gavrila got into a motorboat, which was located on the shore halfway between the first siren and the beginning of the bridge. The experimenters started simultaneously, with the speeds of the bicycle and the motorboat relative to the water being 20 km/h and directed perpendicular to the shore. It turned out that the sound signals from both sirens reached Gavrila simultaneously. Determine the distance from the starting point where Gavrila will be when he is 40 m away from the shore. Round your answer to the nearest whole number of meters. The riverbank is straight, and the current at each point is directed along the shore.
Solution. Let's introduce a coordinate system, with the $x$-axis directed along the shore, and the origin at Gavrila's starting point. The siren on the shore has coordinates $(L, 0), L=50$ m, and Glafira is traveling along the line $x=-L$. Since the experimenters are at the same distance from the shore, the equality of the times needed for the sound signal to travel gives the condition on the coordinates of Gavrila $(x, y)$:

$$
x+L=\sqrt{(x-L)^{2}+y^{2}}
$$

or

$$
y^{2}=4 x L,
$$

from which

$$
x=\frac{y^{2}}{4 L}
$$

which describes a parabola.

According to the problem, the value of $y=40$ m is known. The distance from Gavrila to the starting point can be found using the Pythagorean theorem:

$$
s=\sqrt{x^{2}+y^{2}}=\sqrt{\frac{y^{4}}{16 L^{2}}+y^{2}}=\frac{y \sqrt{y^{2}+16 L^{2}}}{4 L}=8 \sqrt{26}
$$

Since $40.5^{2}<(8 \sqrt{26})^{2}<41^{2}$, the nearest integer number of meters is 41.

Answer 41.