91. Motorboat (I). The smugglers' motorboat has three times the speed of the patrol ship, which is located at a distance of half the way from the boat to the point on the coast that the boat wants to reach. The captain of the boat decides to sail to the goal along two sides of a square. What part of this path will be dangerous?
91. At the moment in question in the problem, the patrol ship is at point $S$, and the motorboat is at point $M$ (Fig. 165). The motorboat moves along the sides of the square $M N, N W$. The patrol ship will not catch up with the boat on the segment $M N$, as its speed is too low for that. Let $a$ be the length of the side of the square, and $v$ be the speed of the ship (thus, the speed of the boat is $3 v$). The point $P$, lying on the segment $N W$, is dangerous for the boat if the patrol ship can reach it no later than the boat, i.e., if

![](https://cdn.mathpix.com/cropped/2024_05_21_5d77f4829f88503a49eag-162.jpg?height=557&width=559&top_left_y=744&top_left_x=1111)

Fig. 165.

$$
\frac{M N + N P}{3 v} \geqslant \frac{S P}{v}
$$

or

$$
a + N P \geqslant 3 \cdot S P, \quad (a + N P)^{2} \geqslant 9(S P)^{2}
$$

Since

$$
(S P)^{2} = (N P)^{2} + \frac{a^{2}}{2} - a \cdot N P
$$

the given inequality is equivalent to the following quadratic inequality with respect to $N P$:

$$
16(N P)^{2} - 22 a \cdot N P + 7 a^{2} \leqslant 0
$$

This inequality holds for

$$
\frac{a}{2} \leqslant N P \leqslant \frac{7 a}{8}
$$

Therefore, the dangerous part of the route is $\frac{3}{8}$ of the total and begins after $\frac{3}{4}$ of the route has been covered, and ends after $\frac{16}{18}$ of the route has been covered.