To compare the volumes of the two cubes and determine the scale factor from the smaller cube to the larger cube, we first establish the volumes given:
- Volume of Cube 1 (smaller cube) = \( V_1 = 512 , \text{m}^3 \)
- Volume of Cube 2 (larger cube) = \( V_2 = 1728 , \text{m}^3 \)
Next, we can find the scale factor between the volumes by using the relationship that for similar shapes (like cubes), the ratio of their volumes is equal to the cube of the ratio of their corresponding side lengths. If \( k \) is the scale factor from the smaller cube to the larger one, then:
\[ \frac{V_2}{V_1} = k^3 \]
Calculating the ratio of the volumes:
\[ \frac{V_2}{V_1} = \frac{1728}{512} \]
Now, simplify this fraction:
\[ \frac{1728 \div 512}{512 \div 512} = \frac{3.375}{1} = 3.375 \]
To express this as a fraction, observe that:
\[ \frac{1728}{512} = \frac{1728 \div 256}{512 \div 256} = \frac{6.75}{2} = \frac{27}{8} \]
This means:
\[ k^3 = \frac{27}{8} \]
Now, 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 \( \frac{3}{2} \).
In summary, the scale factor from the smaller cube to the larger cube is:
\[ \frac{3}{2} \]