To find the surface area of the new cylinder after dilation by a scale factor of \( \frac{1}{2} \), we can use the fact that when an object is dilated by a scale factor \( k \), its surface area is multiplied by \( k^2 \).
In this case, the scale factor \( k \) is \( \frac{1}{2} \). Therefore, the surface area of the new cylinder will be:
\[ \text{New Surface Area} = \text{Original Surface Area} \times k^2 \]
Substituting in the values we have:
\[ \text{New Surface Area} = 42\pi , \text{cm}^2 \times \left(\frac{1}{2}\right)^2 \] \[ = 42\pi , \text{cm}^2 \times \frac{1}{4} \] \[ = \frac{42\pi}{4} , \text{cm}^2 \] \[ = 10.5\pi , \text{cm}^2 \]
Thus, the surface area of the new cylinder is \( 10.5\pi , \text{cm}^2 \).
The correct answer is \( 10.5\pi , \text{cm}^2 \).