To find the scale factor between the two similar cubes based on their volumes, we can use the relationship between the volumes of similar shapes. The volume of a similar shape scales with the cube of the linear dimension scale factor \( k \).
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Finding the scale factor from the volumes:
The formula relating the volume scale factor \( k^3 \) to the volumes \( V_1 \) and \( V_2 \) of two similar objects is: \[ k^3 = \frac{V_2}{V_1} \]
Given:
- Volume of the smaller cube, \( V_1 = 512 , \text{m}^3 \)
- Volume of the larger cube, \( V_2 = 1728 , \text{m}^3 \)
Now, substituting in the values: \[ k^3 = \frac{1728}{512} \]
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Calculating the fraction: \[ k^3 = \frac{1728 \div 512}{512 \div 512} = \frac{3.375}{1} = 3.375 \]
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Finding \( k \): To find \( k \), take the cube root of \( k^3 \): \[ k = \sqrt[3]{\frac{1728}{512}} = \sqrt[3]{\frac{1728 \div 512}{512 \div 512}} = \sqrt[3]{3.375} \] \[ k = \frac{\sqrt[3]{1728}}{\sqrt[3]{512}} = \frac{12}{8} = \frac{3}{2} \]
Therefore, the scale factor from the smaller cube to the larger cube is: \[ \boxed{\frac{3}{2}} \]