The diameter of the base circle of a straight cone and its slant height are both $20 \mathrm{~cm}$. What is the maximum length of a strip that can be attached to the cone's lateral surface without bending, breaking (cutting), or overlapping, if the strip's width is $2 \mathrm{~cm}$?
The diameter of the base circle of the cone is $20 \mathrm{~cm}$, and its circumference is $20 \pi$, which is exactly half the circumference of a circle with a radius of $20 \mathrm{~cm}$. Therefore, when the cone's lateral surface is flattened, it forms a semicircle with a radius of $20 \mathrm{~cm}$. In this semicircle, a $2 \mathrm{~cm}$ wide strip needs to be placed. The longer side of the strip, according to the condition, is a chord of the semicircle. We know that among the chords inscribed in a circle, the longest is the diameter. The longer side of the strip will be the longest when it passes through the center of the semicircle. Using the Pythagorean theorem:

$$
\frac{h}{2}=\sqrt{20^{2}-2^{2}}=\sqrt{396} \approx 19.89
$$

Thus, the length of the strip can be at most $39.7 \mathrm{~cm}$.

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