To compare the volumes of the two cubes and determine the scale factor from the smaller cube to the larger cube, we start by identifying the volumes provided:
- Volume of the smaller cube (V₁) = 512 m³
- Volume of the larger cube (V₂) = 1728 m³
The formula for the volume \( V \) of a cube in terms of its side length \( s \) is:
\[ V = s^3 \]
To find the side lengths of the smaller and larger cubes, we take the cube root of their respective volumes.
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Finding the side length of the smaller cube: \[ s_1 = \sqrt[3]{512} = 8 , \text{m} \]
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Finding the side length of the larger cube: \[ s_2 = \sqrt[3]{1728} = 12 , \text{m} \]
Next, we can determine the scale factor from the smaller cube to the larger cube by forming the ratio of the side lengths:
\[ \text{Scale factor} = \frac{s_2}{s_1} = \frac{12}{8} = \frac{3}{2} \]
To find the relationship between the volumes, note that when two similar cubes have a scale factor of \( k \), the ratio of their volumes is given by \( k^3 \):
\[ \left(\frac{12}{8}\right)^3 = \left(\frac{3}{2}\right)^3 = \frac{27}{8} \]
To summarize:
- The volume of the smaller cube is \( 512 \) m³.
- The volume of the larger cube is \( 1728 \) m³.
- The scale factor from the smaller cube to the larger cube is \( \frac{3}{2} \).
Thus, the scale factor is \( \frac{3}{2} \) as a fraction.
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