To determine the scale factor between the two similar cubes based on their volumes, we will use the formula that relates the volumes of similar shapes.
If the volumes of two similar shapes are \( V_1 \) and \( V_2 \), then the scale factor \( k \) from the smaller shape to the larger shape can be found using the following relationship:
\[ k = \left( \frac{V_2}{V_1} \right)^{\frac{1}{3}} \]
In this case, we have:
- Volume of the smaller cube \( V_1 = 512 , \text{m}^3 \)
- Volume of the larger cube \( V_2 = 1728 , \text{m}^3 \)
Now, plug in the values:
\[ k = \left( \frac{1728}{512} \right)^{\frac{1}{3}} \]
First, calculate the ratio:
\[ \frac{1728}{512} = \frac{1728 \div 512}{512 \div 512} = \frac{3.375}{1} = 3.375 \]
Now, taking the cube root:
\[ k = 3.375^{\frac{1}{3}} = \frac{12}{8} = \frac{3}{2} \]
Thus, the scale factor from the smaller cube to the larger cube, expressed as a fraction, is:
\[ \frac{3}{2} \]
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