To find the surface area of the dilated cube, we first need to determine the surface area of the original cube and then apply the scale factor to find the surface area of the dilated cube.
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Calculate the surface area of the original cube: The formula for the surface area \( A \) of a cube is given by: \[ A = 6 \times s^2 \] where \( s \) is the length of one side of the cube.
For the original cube, with side length \( s = 8 \) cm: \[ A = 6 \times 8^2 = 6 \times 64 = 384 \text{ cm}^2 \]
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Determine the side length of the dilated cube: The scale factor for dilation is 14, so the new side length \( s' \) of the dilated cube will be: \[ s' = 14 \times 8 = 112 \text{ cm} \]
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Calculate the surface area of the dilated cube: Now, we can calculate the surface area of the dilated cube using the new side length: \[ A' = 6 \times (s')^2 = 6 \times (112)^2 \] First, calculate \( (112)^2 \): \[ 112^2 = 12544 \] Now substitute this value back to calculate \( A' \): \[ A' = 6 \times 12544 = 75264 \text{ cm}^2 \]
Therefore, the surface area of the dilated cube is 75264 cm².
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