To find the surface area of the new cylinder after dilation with a scale factor of 1/2, we can use the property that the surface area of a three-dimensional object scales with the square of the scale factor.
Given that the original surface area is \( 42\pi , \text{cm}^2 \), we can find the new surface area as follows:
-
Calculate the square of the scale factor: \[ \left( \frac{1}{2} \right)^2 = \frac{1}{4} \]
-
Multiply the original surface area by this factor: \[ \text{New Surface Area} = \text{Original Surface Area} \times \frac{1}{4} \] \[ \text{New Surface Area} = 42\pi \times \frac{1}{4} = \frac{42\pi}{4} = 10.5\pi, \text{cm}^2 \]
Thus, the surface area of the new cylinder after dilation is \( 10.5\pi , \text{cm}^2 \).
The correct response is: 10.5π cm².