To find the surface area of the dilated cube, you need to follow these steps:
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Calculate the original surface area of the cube: The formula for the surface area \( A \) of a cube is given by:
\[ A = 6s^2 \]
where \( s \) is the length of a side of the cube.
For the original cube with \( s = 8 \) cm:
\[ A = 6(8^2) = 6(64) = 384 \text{ cm}^2 \]
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Determine the new side length after dilation: When a cube is dilated by a scale factor, the new side length \( s' \) can be calculated as:
\[ s' = \text{scale factor} \times s \]
For a scale factor of 14:
\[ s' = 14 \times 8 = 112 \text{ cm} \]
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Calculate the surface area of the dilated cube: Now we use the new side length to calculate the surface area of the dilated cube:
\[ A' = 6(s')^2 = 6(112^2) \]
First, calculate \( 112^2 \):
\[ 112^2 = 12544 \]
Then calculate the surface area:
\[ A' = 6(12544) = 75264 \text{ cm}^2 \]
Therefore, the surface area of the dilated cube is 75264 cm².
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