57. The tape on the tube. 25 m of tape 0.1 mm thick was tightly wound on a cardboard tube - a roll with a diameter of 1 dm was obtained. What is the diameter of the tube?
57. The cross-sectional area of the roller is $25 \pi \mathrm{cm}^{2}$, the tape covers an area of $25 \mathrm{~cm}^{2}$, so the core has an area of $25(\pi-1)$ cm ${ }^{2}$. Let $d$ be the diameter of the tube (without the tape); then from the equation

$$
\pi \frac{d^{2}}{4}=25(\pi-1)
$$

it follows that

$$
d=10 \sqrt{\frac{\pi-1}{\pi}} \mathrm{cm} \approx 8.26 \mathrm{~cm}
$$

![](https://cdn.mathpix.com/cropped/2024_05_21_5d77f4829f88503a49eag-126.jpg?height=594&width=514&top_left_y=1322&top_left_x=1176)

Fig. 127.