To find the surface area of the dilated cube, we first need to establish the surface area of the original cube and then apply the scale factor.
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Determine the original surface area of the cube: The surface area \( (SA) \) of a cube is given by the formula: \[ SA = 6s^2 \] where \( s \) is the side length of the cube.
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Apply the scale factor: If the cube is dilated by a scale factor of \( k \), the new side length will be: \[ s' = k \cdot s \] The surface area of the dilated cube can be calculated using the new side length: \[ SA' = 6(s')^2 = 6(k \cdot s)^2 = 6k^2s^2 \] Therefore, the surface area of the dilated cube is \( k^2 \) times the surface area of the original cube.
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Calculate the surface area of the dilated cube: Given the scale factor \( k = 14 \), the new surface area will be: \[ SA' = 6(14^2) \cdot s^2 \] \[ SA' = 6 \cdot 196 \cdot s^2 \] \[ SA' = 1176 \cdot s^2 \]
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Final Answer: The surface area of the dilated cube is \( 1176 \cdot s^2 \), where \( s^2 \) is the area based on the original cube's side length.
If you have the value for \( s^2 \), you can multiply that by 1176 to get the specific numerical surface area. Otherwise, the surface area of the dilated cube in terms of \( s \) is \( 1176s^2 \) cm².
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