To determine the scale factor from the smaller cube to the larger cube, we first need to find the side lengths of both cubes using their volumes.
The volume \( V \) of a cube with side length \( s \) is given by the formula:
\[ V = s^3 \]
- For the smaller cube with volume \( V = 512 \):
\[ s_1^3 = 512 \]
Taking the cube root of both sides:
\[ s_1 = \sqrt[3]{512} = 8 \]
- For the larger cube with volume \( V = 1728 \):
\[ s_2^3 = 1728 \]
Taking the cube root of both sides:
\[ s_2 = \sqrt[3]{1728} = 12 \]
Now, we can find the scale factor from the smaller cube to the larger cube, which is given by the ratio of their side lengths:
\[ \text{Scale factor} = \frac{s_2}{s_1} = \frac{12}{8} = \frac{3}{2} \]
Next, we can compare their volumes. The volumes of similar shapes have a relationship given by the cube of the scale factor:
\[ \text{Volume ratio} = \left(\text{Scale factor}\right)^3 = \left(\frac{3}{2}\right)^3 = \frac{27}{8} \]
To verify this relation, we can also compare the volumes directly:
\[ \frac{V_2}{V_1} = \frac{1728}{512} = \frac{27}{8} \]
Thus, the volumes of the two cubes are indeed in the ratio of \( \frac{27}{8} \), confirming that the scale factor from the smaller cube to the larger cube is:
\(\frac{3}{2}\).
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