To determine the scale factor between the two boxes, we need to compare their volumes. The volumes given are 648 cubic inches for the first box and 24 cubic inches for the second box.
- Calculate the ratio of the volumes:
\[ \text{Scale Factor} = \frac{\text{Volume of the larger box}}{\text{Volume of the smaller box}} = \frac{648}{24} \]
- Simplify the fraction:
\[ \frac{648}{24} = 27 \]
- The scale factor in terms of volumes is 27.
Now, since the boxes are similar, the scale factor of their linear dimensions (length, width, height) is the cube root of the volume scale factor.
- Calculate the linear scale factor:
\[ \text{Linear Scale Factor} = \sqrt[3]{27} = 3 \]
Thus, the scale factor for the dimensions of the boxes is 3.
In summary, the ratio between the two boxes:
- Volume of larger box = 648 cubic inches
- Volume of smaller box = 24 cubic inches
- Volume Scale Factor = 27
- Linear Scale Factor = 3
The scale factor between the two boxes for their linear dimensions is 3.
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