A cylinder has a surface area of 42 π

To find the surface area of the new cylinder after it is dilated using a scale factor of \( \frac{1}{2} \), we need to use the property of dilation concerning surface area. When a three-dimensional object is dilated by a scale factor \( k \), the new surface area is scaled by a factor of \( k^2 \).

1. Identify the scale factor: \[ k = \frac{1}{2} \]

2. Calculate the new surface area: \[ \text{New Surface Area} = \text{Original Surface Area} \times k^2 \] \[ \text{New Surface Area} = 42\pi \times \left( \frac{1}{2} \right)^2 \] \[ = 42\pi \times \frac{1}{4} \] \[ = \frac{42\pi}{4} \] \[ = 10.5\pi \text{ cm}^2 \]

Thus, the surface area of the new cylinder is \( \boxed{10.5\pi} \) cm squared.