## 16. Math Puzzle $9 / 66$

A level contains a glass tube, called "Dragonfly." Such a glass tube is filled with a liquid, on which a small air bubble floats. There is a marking point on the glass tube. If the level is exactly horizontal, the air bubble aligns with the marking point.

By how many $\mathrm{mm}$ are the marking point and the air bubble apart if the level is tilted by $0.5^{\circ}$?

The radius of curvature of the glass tube is $1 \mathrm{~m}$.
On the periphery of a circle lies the outer side of the dragonfly. The circumference of the circle is $U=2 \pi r=6280 \text{ mm}$. This is the circumference for an angle of $360^{\circ}$.

![](https://cdn.mathpix.com/cropped/2024_06_06_69be9c4355ed0dc2557cg-08.jpg?height=354&width=566&top_left_y=246&top_left_x=745)

The water level is then tangent to the circle. If the water level is tilted by $0.5^{\circ}$, the radius also shifts by $0.5^{\circ}$, because the radius and the tangent form a right angle. The point of contact $P_{1}$ is displaced from point $P_{0}$ (marking on the dragonfly) by:

$$
a=\frac{6280}{360^{\circ}} \cdot 0.5^{\circ}=8 \frac{13}{18} \approx 8.72 \text{ mm}
$$