To compare the volumes of two similar cubes and determine the scale factor from the smaller cube to the larger cube, we need the relationship between the side lengths and volumes of the cubes.
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Scale Factor: If the side length of the smaller cube is \( a \) and the side length of the larger cube is \( b \), the scale factor from the smaller cube to the larger cube is given by: \[ \text{Scale Factor} = \frac{b}{a} \]
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Volume Relationship: The volume \( V \) of a cube is calculated using the formula: \[ V = s^3 \] where \( s \) is the length of a side of the cube. Thus, the volume of the smaller cube is: \[ V_{\text{smaller}} = a^3 \] and the volume of the larger cube is: \[ V_{\text{larger}} = b^3 \]
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Volume Ratio: Since the cubes are similar, the ratio of their volumes is related to the cube of the scale factor: \[ \frac{V_{\text{larger}}}{V_{\text{smaller}}} = \left(\frac{b}{a}\right)^3 \]
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Example Calculation: To perform the calculations, let's say the side length of the smaller cube is \( a = 2 \) and the side length of the larger cube is \( b = 4 \). The scale factor would be: \[ \text{Scale Factor} = \frac{b}{a} = \frac{4}{2} = 2 \]
The volumes would then be: \[ V_{\text{smaller}} = 2^3 = 8 \] \[ V_{\text{larger}} = 4^3 = 64 \] And the volume ratio would be: \[ \frac{V_{\text{larger}}}{V_{\text{smaller}}} = \frac{64}{8} = 8 \] Confirming that: \[ \left(\frac{b}{a}\right)^3 = 2^3 = 8 \]
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Final Result: If your cubes have different specific side lengths, replace \( a \) and \( b \) with the respective values.
In summary, to find the scale factor from the smaller cube to the larger cube, use: \[ \text{Scale Factor} = \frac{\text{Side Length of Larger Cube}}{\text{Side Length of Smaller Cube}} \] Provide your specific side lengths for accurate calculation. If ratios/values are given, substitute them and express the result as a fraction.
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