To find the new surface area after dilation, we first need to understand how the surface area of a solid changes with dilation. When an object is dilated by a scale factor \( k \), the surface area of the object changes by a factor of \( k^2 \).
Here, the original surface area of the cylinder is given as \( 42\pi \) cm², and the scale factor is \( \frac{1}{2} \).
- Calculate the square of the scale factor:
\[ k^2 = \left(\frac{1}{2}\right)^2 = \frac{1}{4} \]
- Now, multiply the original surface area by this factor:
\[ \text{New Surface Area} = \text{Original Surface Area} \times k^2 = 42\pi \times \frac{1}{4} \]
\[ \text{New Surface Area} = \frac{42\pi}{4} = 10.5\pi , \text{cm}^2 \]
Thus, the surface area of the new cylinder after dilation is \( 10.5\pi , \text{cm}^2 \).
The correct answer is 10.5π cm².