Task 6. (30 points) To protect an oil platform located at sea, it is necessary to distribute 8 radars around it, the coverage of each of which is a circle with a radius of $r=15$ 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 18 km. Calculate the area of this ring of coverage.
# Solution.

To ensure radar coverage of a ring around the platform, it is necessary to place them at the vertices of a regular polygon, the center of which coincides with the center of the platform (Fig. 2).

![](https://cdn.mathpix.com/cropped/2024_05_06_b2236406d6db6e8875c1g-29.jpg?height=711&width=734&top_left_y=1592&top_left_x=455)

Fig. 2.

![](https://cdn.mathpix.com/cropped/2024_05_06_b2236406d6db6e8875c1g-29.jpg?height=597&width=509&top_left_y=1706&top_left_x=1256)

Fig. 3.

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 $\triangle B C D$, by the Pythagorean theorem, we find $B D$:

$B D=\sqrt{15^{2}-9^{2}}=12$.

Then $A B=24$, so the distance from the center of the platform to the radars is equal to the radius of the circumscribed circle around the regular octagon:

$$
O B=\frac{A B}{2 \sin (180 / 8)}=\frac{24}{2 \sin 22.5^{\circ}}=\frac{12}{\sin 22.5^{\circ}}
$$

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 $O E$.

$$
\begin{aligned}
& S_{\text {ring }}=\pi\left(O E^{2}-O C^{2}\right) \\
& O D=\frac{B D}{\tan 22.5^{\circ}}=\frac{12}{\tan 22.5^{\circ}} \\
& O C=O D-9=\frac{12}{\tan 22.5^{\circ}}-9 \\
& O E=O D+9=\frac{12}{\tan 22.5^{\circ}}+9 \\
& S_{\text {ring }}=\pi\left(\left(\frac{12}{\tan 22.5^{\circ}}+9\right)^{2}-\left(\frac{12}{\tan 22.5^{\circ}}-9\right)^{2}\right)=\frac{432 \pi}{\tan 22.5^{\circ}}
\end{aligned}
$$