Solution: The rolled-up ribbon

We now publish the solution to the divertimento The rolled-up ribbon.

Divertimento:

We wind a ribbon 25 m long and 0.1 mm thick around a cardboard cylinder, obtaining a cylindrical roll whose diameter is 1 dm.

What is the diameter of the cardboard cylinder?

Solution:

The surface area of the roll section is
$$
\pi \cdot \Big(\frac{1}{2}\Big)^2\, \text{dm}^2 = 25 \pi\,\text{cm}^2.
$$
In this section, the rolled-up loop occupies a surface area equal to
$$
250 \cdot 0.01=25\,\text{cm}^2.
$$
Therefore, the surface area of the cardboard cylinder is
$$
25 \pi – 25 = 25(\pi-1) \, \text{cm}^2.
$$

If we call $\(d\) the diameter of the cardboard cylinder in centimetres, we have that
$$\pi \frac{d^2}{4} = 25(\pi -1),$$
therefore
$$
d=10 \sqrt{\frac{\pi-1}{\pi}} \, \text{cm}.
$$