The spider can reach the point on the floor opposite the edge of the ceiling of a cylindrical room in two ways, covering the same distance:

1. It slides down vertically, then continues its journey along the diameter of the floor.
2. It travels along the surface of the cylinder. When the surface is laid flat, its path is a straight line.

Calculate the ratio of the height of the cylinder to its diameter.
Let's denote the height of the cylinder by $m$ and the radius of its base circle by $r$. The distance traveled in the first path is: $m + 2r$.

Unroll the cylinder's lateral surface onto a plane and draw the second path. This is the hypotenuse of a right triangle with legs $m$ and $r\pi$. By the Pythagorean theorem, it is $\sqrt{m^{2} + (r\pi)^{2}}$. From the equality of the two paths,

$$
m + 2r = \sqrt{m^{2} + (r\pi)^{2}}
$$

Rearrange the equation; we get

$$
4mr = r^{2}(\pi^{2} - 4)
$$

Dividing by $8r^{2} \neq 0$, the desired ratio is

$$
\frac{m}{2r} = \frac{\pi^{2} - 4}{8} \approx 0.7337
$$

![](https://cdn.mathpix.com/cropped/2024_05_02_c6f44018fa358daf72f3g-1.jpg?height=575&width=935&top_left_y=683&top_left_x=584)