To determine whether the two figures are similar based on their volumes, we need to know the specific volumes of each figure and the relationship between these volumes.
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Understanding Similar Figures:
- Two figures are similar if their corresponding dimensions (lengths, heights, etc.) are in proportion.
- The volume ratio of similar figures is equal to the cube of the ratio of their corresponding linear dimensions.
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Volume Ratios:
- If the volume of the larger figure is \( V_1 \) and that of the smaller figure is \( V_2 \), then the ratio of their volumes is \( \frac{V_1}{V_2} \).
- If the ratio of the volume is \( 125 : 1 \), then the ratio of their corresponding linear dimensions is \( \sqrt[3]{125} : 1 \), which simplifies to \( 5 : 1 \).
- Therefore, if the volume ratio is \( 125 : 1 \), the linear dimension ratio is \( 5 : 1 \).
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Identifying the Correct Statement:
- If the figures have a volume ratio of \( 125 : 1 \), they are similar, as their linear dimensions have a ratio of \( 5 : 1 \).
- The other options suggest that the figures do not maintain similarity based on equal volumes or lack of a stated scale factor, both of which are incorrect if the ratio is \( 125 : 1 \).
Based on these points, the correct statement would be: The two figures are similar because the volumes have a ratio of 125 : 1.