To find the surface area of cube B, which is the image of cube A after dilation by a scale factor of 4, we can use the property of surface area in relation to the scale factor.
When a three-dimensional shape such as a cube is dilated by a scale factor \( k \), the surface area of the new shape increases by a factor of \( k^2 \).
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Calculate the scale factor: The scale factor \( k \) is 4.
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Find the square of the scale factor: \[ k^2 = 4^2 = 16 \]
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Calculate the surface area of cube B: Since the surface area of cube A is given as 114 inĀ², we can find the surface area of cube B by multiplying the surface area of cube A by \( k^2 \): \[ \text{Surface area of cube B} = \text{Surface area of cube A} \times k^2 \] \[ \text{Surface area of cube B} = 114 , \text{in}^2 \times 16 = 1824 , \text{in}^2 \]
Therefore, the surface area of cube B is \( 1824 , \text{in}^2 \).