The surface area \( S \) of a cylinder can be expressed as:
\[ S = 2\pi rh + 2\pi r^2 \]
where \( r \) is the radius and \( h \) is the height of the cylinder.
When a cylinder is dilated by a scale factor of \( k \), the new dimensions (both the radius and the height) are multiplied by \( k \). This means that if the original radius is \( r \) and the original height is \( h \), after dilation by a scale factor of \( k \):
\[ \text{New radius} = kr \ \text{New height} = kh \]
The surface area of the new cylinder \( S' \) becomes:
\[ S' = 2\pi (kr)(kh) + 2\pi (kr)^2 \] \[ = 2\pi k^2 rh + 2\pi k^2 r^2 \] \[ = k^2 (2\pi rh + 2\pi r^2) \] \[ = k^2 S \]
Given that the original surface area \( S \) is \( 42\pi \) cm² and the scale factor \( k \) is \( \frac{1}{2} \):
Calculate the new surface area:
\[ S' = \left(\frac{1}{2}\right)^2 S \ = \frac{1}{4} S \ = \frac{1}{4} \times 42\pi \ = \frac{42}{4}\pi \ = 10.5\pi \text{ cm}^2 \]
Therefore, the surface area of the new cylinder after dilation is \( 10.5\pi \) cm².
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