To solve for the new surface area of the dilated cylinder, we need to understand how surface area is affected by dilation.
When an object is dilated by a scale factor \( k \), the surface area of the object is affected by this factor squared. In this case, the scale factor is \( \frac{1}{2} \).
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Calculate the new scale factor for surface area: \[ \text{New Surface Area} = \text{Original Surface Area} \times \left(\text{Scale Factor}\right)^2 \]
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Apply the original surface area, which is \( 54\pi , \text{cm}^2 \): \[ \text{New Surface Area} = 54\pi \times \left(\frac{1}{2}\right)^2 \] \[ \left(\frac{1}{2}\right)^2 = \frac{1}{4} \] Therefore, \[ \text{New Surface Area} = 54\pi \times \frac{1}{4} = \frac{54\pi}{4} = 13.5\pi , \text{cm}^2 \]
So, the surface area of the new dilated cylinder is:
D. 13.5π cm²