5) An atoll has the shape of a circular crown delimited by two concentric circumferences with radii of $1 \mathrm{~km}$ and $6 \mathrm{~km}$, respectively. Giovanni and Marco are the only inhabitants of the atoll; after a storm that destroyed their huts, they decide to rebuild them as far apart as possible, but in such a way that there exists a straight path connecting them and lying entirely on the atoll. How far apart will Giovanni's and Marco's huts be? (Assume that the two circumferences delimiting the atoll are part of it)
(A) $\frac{\sqrt{35}}{2} \mathrm{~km}$,
(B) $\frac{\sqrt{37}}{2} \mathrm{~km}$,
(C) $\sqrt{37} \mathrm{~km}$,
(D) $2 \sqrt{35} \mathrm{~km}$,
(E) $2 \sqrt{37} \mathrm{~km}$.
5. The answer is (D). The maximum possible distance between the two huts, consistent with the fact that there exists a straight path entirely contained within the atoll that connects them, is the length of a chord of the circumference that delimits the atoll externally, tangent to the circumference that delimits it internally. A chord of this type is the segment $A C$ in the figure below. The length of $A C$ is twice the length of $B C$; this can be found by applying the Pythagorean Theorem to the triangle $B O C: \overline{B C}^{2}=\overline{O C}^{2}-\overline{O B}^{2}=35 \mathrm{~km}$. Therefore, $\overline{A C}=2 \sqrt{35} \mathrm{~km}$.

![](https://cdn.mathpix.com/cropped/2024_04_17_d9e56cae5925cb0c0feag-2.jpg?height=489&width=551&top_left_y=518&top_left_x=798)