92. Motorboat (II). In the previous problem, the captain of the boat decided to sail in such a way as to change course by $90^{\circ}$ once along the way. What route should he choose to definitely avoid the patrol ship and reach his destination as quickly as possible?

## $\Gamma$ L A B A VI

## MATHEMATICAL ADVENTURES OF DOCTOR SHARADEK
92. At the initial moment, the boat is at point $M$, the patrol ship is at point $S$; the boat must reach the shore at point $W$ (Fig. 166). Points $M, S, W$ lie on the same line $M S \equiv S W$. From the text of the problem, it follows that the boat must sail along the sides of the right angle $M N W$, inscribed in a circle with diameter $M W$.

The path of the boat should be as short as possible, so the first segment of it ($M N$) should form as small an angle as possible with the direction $M W$. In addition, the boat must certainly avoid the patrol ship. Therefore, its path should not intersect the Apollonian circle $T U V$, which is the locus of points where

![](https://cdn.mathpix.com/cropped/2024_05_21_5d77f4829f88503a49eag-163.jpg?height=637&width=520&top_left_y=475&top_left_x=397)

Fig. 166. the ship and the boat, sailing from points $S$ and $M$ respectively at their maximum speeds along straight lines. The center of the Apollonian circle is point $O$, whose position on the segment $M W$ is determined from the relations

$$
\begin{gathered}
M T=3 S T \\
M V=3 S V \\
O T=O V
\end{gathered}
$$

From these equalities, it follows that

$$
M O=\frac{9}{16} M W
$$

and the radius of the Apollonian circle is $\frac{3}{16} M W$. If the segment $M N$ were tangent to the circle $T U V$, then the path of the boat would be $\frac{2 \sqrt{2}+1}{3} M W$, and if $M N$ does not have common points with this circle, then the path will be longer than $\frac{2 \sqrt{2}+1}{3} M W$. It remains to check that if the segment $M N$ passes sufficiently close to the circle $T U V$, the patrol ship will not catch the boat on the segment $N W$. We leave this to the readers.

Thus, the course of the boat, satisfying the conditions of the problem, is fully determined.