To find the surface area of the new cylinder after dilating it with a scale factor of \( \frac{1}{2} \), we need to understand how surface area changes with dilation.
When an object is dilated by a scale factor \( k \), the surface area changes by a factor of \( k^2 \). In this case, the scale factor is \( \frac{1}{2} \).
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Calculate the factor by which the surface area changes: \[ k^2 = \left( \frac{1}{2} \right)^2 = \frac{1}{4} \]
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Now, take the original surface area and multiply it by this factor: \[ \text{New Surface Area} = \text{Original Surface Area} \times \frac{1}{4} \] \[ \text{New Surface Area} = 42 \pi , \text{cm}^2 \times \frac{1}{4} = \frac{42 \pi}{4} , \text{cm}^2 = 10.5 \pi , \text{cm}^2 \]
Therefore, the surface area of the new cylinder is:
\[ \boxed{10.5 \pi , \text{cm}^2} \]
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