Task 6. (30 points) To protect an oil platform located at sea, it is necessary to distribute 7 radars around it, the coverage of each of which is a circle with a radius of $r=41$ 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, 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 $\triangle B C D$, by the Pythagorean theorem, we find $B D$:

$B D=\sqrt{41^{2}-9^{2}}=40$.

Then $A B=80$, so the distance from the center of the platform to the radars is equal to the radius of the circumscribed circle around a regular heptagon:

$$
O B=\frac{A B}{2 \sin (180 / 7)}=\frac{80}{2 \sin (180 / 7)}=\frac{40}{\sin (180 / 7)}
$$

![](https://cdn.mathpix.com/cropped/2024_05_06_b2236406d6db6e8875c1g-15.jpg?height=736&width=760&top_left_y=223&top_left_x=428)

Fig. 2.

![](https://cdn.mathpix.com/cropped/2024_05_06_b2236406d6db6e8875c1g-15.jpg?height=625&width=525&top_left_y=350&top_left_x=1254)

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 $O E$.

$S_{\text {ring }}=\pi\left(O E^{2}-O C^{2}\right)$.

$O D=\frac{B D}{\operatorname{tg}(180 / 7)}=\frac{40}{\operatorname{tg}(180 / 7)} ;$

$O C=O D-9=\frac{40}{\operatorname{tg}(180 / 7)}-9$;

$O E=O D+9=\frac{40}{\operatorname{tg}(180 / 7)}+9$;

$S_{\text {ring }}=\pi\left(\left(\frac{40}{\operatorname{tg}(180 / 7)}+9\right)^{2}-\left(\frac{40}{\operatorname{tg}(180 / 7)}-9\right)^{2}\right)=\frac{1440 \pi}{\operatorname{tg}(180 / 7)}$.

Answer. $\frac{40}{\sin (180 / 7)} ; \frac{1440 \pi}{\operatorname{tg}(180 / 7)}$.

## Variant-22