The thickness of a cylinder solid is 0.8 cm. If 10 such pieces of cylinder solid are piled on top of the other. Find the limits within which the heights of the pile lie.

Given that the thickness of each cylindrical solid is 0.8 cm, and there are 10 such cylindrical solids piled on top of each other, we need to find the limits within which the height of the pile lies.

### Calculations

1. **Minimum Height**: If the cylinders are piled perfectly one on top of the other with no gaps, the minimum height \(H_{min}\) of the pile is given by:
   \[
   H_{min} = 10 \times 0.8 \, \text{cm} = 8 \, \text{cm}
   \]

2. **Maximum Height**: If there are slight gaps or misalignments between the cylinders, the maximum height \(H_{max}\) would depend on the extent of these gaps. However, without additional information about the maximum gap possible, it’s reasonable to assume the maximum height would slightly exceed the minimum height due to such imperfections.

Since no specific gap is provided, the problem is typically interpreted as the height being \(8 \, \text{cm}\) (as the pile is assumed to be perfectly aligned). If there were additional details like potential gaps or misalignment, the upper limit could be adjusted accordingly.

### Conclusion

Therefore, under the assumption of perfect alignment, the height of the pile lies within the limits:

\[
\boxed{8 \, \text{cm}}
\]

If you assume there might be small gaps, the height could be slightly more, but without specific information on the gap size, it's best to stick with the precise calculation.

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Would you like more details or have any questions about this calculation?

### Follow-Up Questions:
1. What would happen if there were small gaps between the cylinders? How would that affect the height?
2. How would the calculation change if the number of cylinders were different?
3. Can you calculate the volume of a single cylinder if its diameter and height are provided?
4. How does the height change if the thickness of the cylinders is doubled?
5. What is the effect on the pile height if the cylinders are stacked with a slight tilt?

### Tip:
When stacking objects like cylinders, always consider possible gaps or misalignments, as they can significantly affect the total height.

Learn how to calculate the height of a pile of cylindrical solids with a detailed step-by-step solution. This problem involves geometric concepts such as measurement and stacking of cylindrical shapes. Suitable for students in Grades 7-9.

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