To compare the volumes of the two cubes, we first note their given volumes:
- Volume of the smaller cube, \( V_1 = 512 , m^3 \)
- Volume of the larger cube, \( V_2 = 1728 , m^3 \)
To determine the scale factor from the smaller cube to the larger cube in terms of volume, we can use the relationship between the volumes and the scale factor. If two similar shapes (in this case, cubes) have a scale factor of \( k \), the ratio of their volumes is given by:
\[ \frac{V_2}{V_1} = k^3 \]
We can rearrange this formula to solve for \( k \):
\[ k = \sqrt[3]{\frac{V_2}{V_1}} \]
First, we calculate the ratio of the volumes:
\[ \frac{V_2}{V_1} = \frac{1728 , m^3}{512 , m^3} = \frac{1728}{512} \]
Now, we simplify \( \frac{1728}{512} \).
Both numbers can be divided by 64:
\[ 1728 \div 64 = 27, \quad 512 \div 64 = 8 \]
So,
\[ \frac{1728}{512} = \frac{27}{8} \]
Now, we can find the scale factor \( k \):
\[ k = \sqrt[3]{\frac{27}{8}} = \frac{\sqrt[3]{27}}{\sqrt[3]{8}} = \frac{3}{2} \]
Thus, the scale factor from the smaller cube to the larger cube is:
\[ \frac{3}{2} \]
In conclusion, the scale factor is \( \frac{3}{2} \).