To compare the volumes of two similar cubes and determine the scale factor from the smaller cube to the larger cube, we need to understand the relationship between the side lengths of the cubes and their volumes.
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Understanding Similar Cubes:
- If two cubes are similar, their corresponding side lengths (edges) are in a specific ratio, which is called the scale factor.
- Let's denote the side length of the smaller cube as \( s_1 \) and the side length of the larger cube as \( s_2 \). The scale factor from the smaller cube to the larger cube is \( \frac{s_2}{s_1} \).
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Volume Calculation:
- The volume \( V \) of a cube is calculated using the formula \( V = s^3 \), where \( s \) is the length of a side.
- Therefore, the volume of the smaller cube \( V_1 \) is \( V_1 = s_1^3 \) and the volume of the larger cube \( V_2 \) is \( V_2 = s_2^3 \).
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Volume Ratio:
- The ratio of the volumes of the two cubes can be expressed as: \[ \frac{V_2}{V_1} = \frac{s_2^3}{s_1^3} = \left(\frac{s_2}{s_1}\right)^3 \]
- This means that the ratio of the volumes of the two cubes is equal to the cube of the ratio of their side lengths (scale factor).
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Finding the Scale Factor:
- If we know either the side lengths or the volumes of the cubes, we can find the scale factor. If for example, we know the side lengths \( s_1 \) and \( s_2 \), the scale factor is simply \( \frac{s_2}{s_1} \).
- Conversely, if we have the volumes \( V_1 \) and \( V_2 \), we can compute the scale factor from the volumes: \[ \text{Scale factor} = \sqrt[3]{\frac{V_2}{V_1}} \]
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Example:
- If the smaller cube has a side length of 2 cm (thus \( V_1 = 2^3 = 8 \) cm³) and the larger cube has a side length of 4 cm (thus \( V_2 = 4^3 = 64 \) cm³), then: \[ \text{Scale factor} = \frac{s_2}{s_1} = \frac{4}{2} = 2 \]
- The volume ratio would be \( \frac{V_2}{V_1} = \frac{64}{8} = 8 \), which equals \( 2^3 \), confirming that the scale factor correctly reflects the relationship between the two volumes.
In summary, knowing the volumes or the side lengths of the cubes allows you to determine the scale factor and compare their volumes as described.
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