In certain cassette tapes, a $0,0075 \mathrm{~mm}$ thick tape is wound on spools with a diameter of $11 \mathrm{~mm}$. The distance between the centers of the spools is $42 \mathrm{~mm}$. What is the maximum length of tape that can be in a usable cassette? (From the full spool, the tape must pass freely to the empty spool.)
The belt rolls up in a cylindrical shape onto the spools, and its length is influenced only by the radii of the spools, apart from the thickness of the belt.

The length of the belt is greatest when the cylinders touch, that is, when the sum of their base areas is the largest. In the case of maximum area, at a certain position, the cylinders just touch each other. Due to symmetry, this holds when the radii of the two cylinders are equal: if the two radii are not equal, then the belt unwound from the larger radius will create a larger radius increase on the smaller radius (with equal belt length), than the decrease in the radius of the other spool. By continuing to wind, we could increase the length of the belt.

When the spools touch, the radius is $21 \mathrm{~mm}$, and the combined area of the circles is $2 \cdot 21^{2} \pi$. To get the length of the belt, we need to subtract the area of the spools from this and divide by the thickness of the belt.

That is, $h=\frac{2 \pi 21^{2}-2 \pi 5.5^{2}}{0.0075} \approx 344109 \mathrm{~mm}$.

Thus, we can use approximately 344 meters of belt.

Széll Károly (Szeged, Ságvári E. Gymnasium, 9th grade)