Is the beach of Balatonmáriafürdő visible from the hill of Balatonvilágos? Is the Nagy Szilfa of Balatonakarattya visible from the boat station of Balatonszemes? - The height of Lake Balaton above sea level is $104 \mathrm{~m}$, let's take the height of the mentioned hill and tree above sea level as $154 \mathrm{~m}$; the two distances on a $1: 95000$ scale map are $70 \mathrm{~cm}$ and $38.5 \mathrm{~cm}$. (Consider the Earth as a sphere, the circumference of its equator is $40 \cdot 10^{3} \mathrm{~km}$. .)
We will receive an answer to both questions, in accordance with the following: when we ascend to a height of $h(=50 \mathrm{~m})$ above the level of Lake Balaton, how far away is the point on the shore that we can just see?

Let the radius of the spherical surface passing through the center $O$ of the Earth and the level of Lake Balaton be $R$, our position be $A$, the projection of $A$ on the spherical surface (the level of Lake Balaton) be $B$, and the point of interest on the shore be $P$. Clearly, $\angle APO = 90^{\circ}$, and by the Pythagorean theorem,

$$
A P^{2}=A O^{2}-O P^{2}, \quad d^{2}=(R+h)^{2}-R^{2}=h(2 R+h)
$$

$R$ is given by the radius of the Earth's main circumference $40 \cdot 10^{3} / 2 \pi$ plus the height $m$ above sea level. For our purposes, it is sufficient to calculate with an accuracy of three significant figures, and thus use the common approximate value of $\pi$ as 3.14. In this case, we also round the above ratio to 6370 km, which is justified by the approximation of the Earth as a sphere. Therefore, there is no point in considering the addition of $m \approx 0.1 \mathrm{~km}$ and $h$ as an addend, so $d^{2} \approx 2 R \cdot h$. In our case, $h=0.05 \mathrm{~km}$, $d=\sqrt{637} \approx 25.2 \mathrm{~km}$.

![](https://cdn.mathpix.com/cropped/2024_05_02_7bf5bf65005f97038131g-1.jpg?height=433&width=1122&top_left_y=659&top_left_x=491)

On the other hand, according to the data on the map, the distances in question are $66.5 \mathrm{~km}$ and $36.6 \mathrm{~km}$, both larger than $d$, so the answer to both questions is: not visible.

Pál Tamási (Győr, Czuczor G. Benedek Secondary School II. o. t.)