To compare the volumes of the two similar cubes and determine the scale factor from the smaller cube to the larger cube, we can start by finding the side lengths of each cube, since the volume \( V \) of a cube with side length \( s \) is given by the formula:
\[ V = s^3 \]
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Finding the side length of the smaller cube: Given the volume of the smaller cube is \( 512 , m^3 \): \[ s_1^3 = 512 \implies s_1 = \sqrt[3]{512} = 8 , m \]
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Finding the side length of the larger cube: Given the volume of the larger cube is \( 1728 , m^3 \): \[ s_2^3 = 1728 \implies s_2 = \sqrt[3]{1728} = 12 , m \]
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Calculating the scale factor: The scale factor from the smaller cube to the larger cube is the ratio of their side lengths: \[ \text{Scale Factor} = \frac{s_2}{s_1} = \frac{12}{8} = \frac{3}{2} \]
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Comparing the volumes: The volume ratio of two similar shapes is the cube of the scale factor. To verify: \[ \left(\frac{3}{2}\right)^3 = \frac{27}{8} \]
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Checking the ratio of the volumes: The volume of the smaller cube is \( 512 , m^3 \) and the volume of the larger cube is \( 1728 , m^3 \): \[ \text{Volume Ratio} = \frac{1728}{512} = \frac{27}{8} \]
Thus, the scale factor from the smaller cube to the larger cube is:
\[ \text{Scale Factor} = \frac{3}{2} \]
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