To compare the volumes of the two cubes, we start with the given volumes:
- Volume of the smaller cube, \( V_1 = 512 , \text{m}^3 \)
- Volume of the larger cube, \( V_2 = 1728 , \text{m}^3 \)
The scale factor for similar shapes (in this case, cubes) can be determined using the relationship between their volumes and their sides. Since the cubes are similar in shape, the ratio of their volumes is equal to the cube of the ratio of their corresponding side lengths (scale factor).
First, we calculate the ratio of the volumes:
\[ \text{Ratio of volumes} = \frac{V_2}{V_1} = \frac{1728}{512} \]
Now, we simplify this fraction:
\[ \frac{1728}{512} = \frac{1728 \div 512}{512 \div 512} = \frac{3.375}{1} = \frac{27}{8} \]
Thus, the ratio of the volumes is \(\frac{27}{8}\).
Since this ratio is equal to the cube of the scale factor \( k \) (where \( k \) is the scale factor from the smaller cube to the larger cube), we have:
\[ k^3 = \frac{27}{8} \]
To find \( k \), we take the cube root of both sides:
\[ k = \sqrt[3]{\frac{27}{8}} = \frac{\sqrt[3]{27}}{\sqrt[3]{8}} = \frac{3}{2} \]
Thus, the scale factor from the smaller cube to the larger cube is:
\[ \frac{3}{2} \]
Final Answer:
The scale factor from the smaller cube to the larger cube is \(\frac{3}{2}\).
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