To determine the scale factor between the two similar cubes based on their volumes, we first need to find the volume of each cube in a consistent unit.
The volumes given are:
- 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 from the smaller cube to the larger cube can be calculated using the formula related to the volumes of similar shapes. If the scale factor is \( k \), then the relationship between the volumes of similar shapes is given by:
\[ \frac{V_2}{V_1} = k^3 \]
Now we can find the scale factor:
\[ k^3 = \frac{V_2}{V_1} = \frac{1728}{512} \]
Simplifying the fraction:
\[ \frac{1728 \div 512}{512 \div 512} = \frac{3.375}{1} = \frac{27}{8} \]
Now, we take the cube root of both sides to find \( k \):
\[ k = \sqrt[3]{\frac{27}{8}} = \frac{\sqrt[3]{27}}{\sqrt[3]{8}} = \frac{3}{2} \]
Thus, the scale factor from the smaller cube to the larger cube is:
\[ \frac{3}{2} \]
So the final answer is:
\[ \frac{3}{2} \]