To find the surface area of a dilated cube, we need to consider how dilation affects both the dimensions and the surface area of the cube.
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Original Surface Area of the Cube: If the original side length of the cube is \( s \), the surface area \( A \) of the cube is calculated using the formula: \[ A = 6s^2 \]
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Dilated Cube: If the cube is dilated by a scale factor of \( k = \frac{1}{4} \), the new side length of the cube \( s'\) will be: \[ s' = k \cdot s = \frac{1}{4}s \]
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New Surface Area: We can calculate the new surface area \( A' \) using the dilated side length: \[ A' = 6(s')^2 = 6\left(\frac{1}{4}s\right)^2 \] \[ A' = 6\left(\frac{1}{16}s^2\right) = \frac{6}{16}s^2 = \frac{3}{8}s^2 \]
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Ratio of Surface Areas: The new surface area compared to the original surface area is: \[ \frac{A'}{A} = \frac{\frac{3}{8}s^2}{6s^2} = \frac{3}{48} = \frac{1}{16} \]
Thus, when a cube is dilated by a scale factor of \( \frac{1}{4} \), the surface area scales down by a factor of \( \frac{1}{16} \).
In conclusion, if you start with a cube that has a certain surface area, after dilation by a factor of \( \frac{1}{4} \), the surface area will be \( \frac{1}{16} \) of the original surface area.