To find the surface area of the new cylinder after it has been dilated using a scale factor of \( \frac{1}{2} \), we can use the property that the surface area of a three-dimensional shape changes with the square of the scale factor.
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Original Surface Area: The original surface area of the cylinder is \( 54\pi \) cm².
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Scale Factor: The scale factor for the dilation is \( \frac{1}{2} \).
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Change in Surface Area: The new surface area after dilation can be calculated using the formula: \[ \text{New Surface Area} = (\text{Scale Factor})^2 \times \text{Original Surface Area} \]
Substituting the values: \[ \text{New Surface Area} = \left(\frac{1}{2}\right)^2 \times 54\pi = \frac{1}{4} \times 54\pi = \frac{54\pi}{4} = 13.5\pi \text{ cm}^2 \]
So, the surface area of the new dilated cylinder is \( 13.5\pi \) cm². The correct answer is indeed:
13.5π cm²
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