We have a brand new pencil. We sharpen the pencil. When the tip wears down, we always sharpen it again, until the length of the pencil decreases to one-tenth of its original length. Estimate what percentage of the pencil lead is lost during the sharpening process.
Every time we sharpen the pencil, we carve a small cylindrical part of the pencil lead into a conical shape. The radius and height of the base circle of the cylinder and the cone are equal, so the volume of the cone is $\frac{1}{3}$ of the volume of the cylinder. Therefore, with each sharpening, $\frac{2}{3}$ of the pencil lead is wasted. If we continue this until the length of the pencil is reduced to $\frac{1}{10}$ of its original length, the total waste is $\frac{9}{10} \cdot \frac{2}{3}=\frac{3}{5}$ of the lead, which means $60\%$ of the lead is wasted.