To find the surface area of the dilated cube, we need to follow these steps:
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Calculate the surface area of the original cube. The surface area \( A \) of a cube is given by the formula:
\[ A = 6s^2 \]
where \( s \) is the length of a side of the cube. For the original cube, the side length is 8 cm.
\[ A = 6 \times (8 , \text{cm})^2 = 6 \times 64 , \text{cm}^2 = 384 , \text{cm}^2 \]
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Determine the new side length after dilation. The cube is dilated by a scale factor of \( \frac{1}{4} \). Therefore, the new side length \( s' \) is:
\[ s' = s \times \frac{1}{4} = 8 , \text{cm} \times \frac{1}{4} = 2 , \text{cm} \]
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Calculate the surface area of the dilated cube. Now, we can find the surface area of the dilated cube using the new side length:
\[ A' = 6 \times (s')^2 = 6 \times (2 , \text{cm})^2 = 6 \times 4 , \text{cm}^2 = 24 , \text{cm}^2 \]
Thus, the surface area of the dilated cube is \( \boxed{24 , \text{cm}^2} \).
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