To determine the scale factor from the smaller cube to the larger cube, we first need to find the ratio of their volumes.
The volume of cube one, \( V_1 = 512 , \text{m}^3 \).
The volume of cube two, \( V_2 = 1728 , \text{m}^3 \).
The ratio of the volumes of the two cubes is:
\[ \text{Volume Ratio} = \frac{V_2}{V_1} = \frac{1728}{512} \]
Now, we can simplify this fraction:
\[ \frac{1728 \div 512}{512 \div 512} = \frac{3.375}{1} = \frac{27}{8} \]
The volumes of similar shapes relate to the scale factor (k) by the formula:
\[ \left(\text{Scale Factor}\right)^3 = \frac{V_2}{V_1} \]
So, we have:
\[ k^3 = \frac{27}{8} \]
Taking the cube root of both sides to 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 cube one to cube two is:
\[ \frac{3}{2} \]
Final Answer: \(\frac{3}{2}\)
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