To find the difference in the lengths of the diagonals of the two tiles, we need to subtract the length of the diagonal of the second tile from the length of the diagonal of the first tile.
The lengths of the diagonals are given as:
- First tile: \( \sqrt{92} \)
- Second tile: \( \sqrt{82} \)
The difference in the lengths of the diagonals can be expressed as:
\[ \sqrt{92} - \sqrt{82} \]
Next, we can simplify this expression further. To find a numerical value, we can calculate \( \sqrt{92} \) and \( \sqrt{82} \):
Calculating \( \sqrt{92} \): \[ \sqrt{92} \approx 9.591663 \]
Calculating \( \sqrt{82} \): \[ \sqrt{82} \approx 9.055385 \]
Now, subtract the two results:
\[ \sqrt{92} - \sqrt{82} \approx 9.591663 - 9.055385 \approx 0.536278 \]
Therefore, the difference in the lengths of the diagonals of the two tiles is approximately:
\[ \boxed{0.536} \] centimeters.
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