Task 4. (20 points) To protect an oil platform located at sea, it is necessary to distribute 9 radars around it, each with a coverage radius of $r=37$ km. Determine the maximum distance from the center of the platform at which they should be placed to ensure that the radars cover a ring around the platform with a width of 24 km. Calculate the area of this ring of coverage.
# Solution.

To ensure radar coverage of a ring around the platform, they need to be placed at the vertices of a regular polygon, the center of which coincides with the center of the platform (Fig. 2).

Point $O$ is the center of the oil platform, and points $A$ and $B$ are the locations of the radars. The circles represent the radar coverage. Consider a fragment of Fig. 2 - triangle $\triangle A O B$ (Fig. 3). In the right triangle $\triangle B C D$, using the Pythagorean theorem, we find $B D$:

$B D=\sqrt{37^{2}-12^{2}}=35$

Then $A B=70$, so the distance from the center of the platform to the radars is equal to the radius of the circumscribed circle around a regular nonagon:

$$
O B=\frac{A B}{2 \sin (180 / 9)}=\frac{70}{2 \sin 20^{\circ}}=\frac{35}{\sin 20^{\circ}}
$$

![](https://cdn.mathpix.com/cropped/2024_05_06_424f80418dd801e5b28ag-44.jpg?height=717&width=716&top_left_y=230&top_left_x=450)

Fig. 2.

![](https://cdn.mathpix.com/cropped/2024_05_06_424f80418dd801e5b28ag-44.jpg?height=597&width=511&top_left_y=341&top_left_x=1258)

Fig. 3.

To find the area of the ring coverage, we need to subtract the area of the circle with radius $O C$ from the area of the circle with radius $O E$.

$$
\begin{aligned}
& S_{\text {ring }}=\pi\left(O E^{2}-O C^{2}\right) \\
& O D=\frac{B D}{\tan 20^{\circ}}=\frac{35}{\tan 20^{\circ}} \\
& O C=O D-12=\frac{35}{\tan 20^{\circ}}-12 \\
& O E=O D+12=\frac{35}{\tan 20^{\circ}}+12 \\
& S_{\text {ring }}=\pi\left(\left(\frac{35}{\tan 20^{\circ}}+12\right)^{2}-\left(\frac{35}{\tan 20^{\circ}}-12\right)^{2}\right)=\frac{1680 \pi}{\tan 20^{\circ}}
\end{aligned}
$$

Answer. $\frac{35}{\sin 20^{\circ}} ; \frac{1680 \pi}{\tan 20^{\circ}}$.