25. Length of a spiral. A piece of wire is wound in a spiral around a cylindrical tube, forming 10 turns. The length of the tube is 9 cm, and the length of its outer circumference is 4 cm. The ends of the spiral lie on the same generatrix of the cylinder. Find the length of the wire.

![](https://cdn.mathpix.com/cropped/2024_05_21_2b40d984baf9da275c68g-017.jpg?height=183&width=808&top_left_y=551&top_left_x=630)
25. Let's unroll the surface of the cylinder along with the wire onto a plane. The generatrix (9 cm), the tenfold repeated circumference ($10 \cdot 4$ cm), and the wire ($L$) now form a right triangle. Therefore, $L=(81+1600)^{\frac{1}{2}}=41$ cm.

![](https://cdn.mathpix.com/cropped/2024_05_21_2b40d984baf9da275c68g-083.jpg?height=160&width=462&top_left_y=922&top_left_x=797)

From this, it follows, among other things, that if the wire is not wound around the cylinder in such a way that it intersects the generatrices at a constant angle, it may loosen over time.

[M. M., 23, 278 (May 1950).]