Question

A cylinder has a surface area of 54π cm2
. If it were dilated using a scale factor of 1/2
, what is the surface area of the new cylinder?(1 point)
Responses

27π cm2
27 pi cm squared

108π cm2
108 pi cm squared

6.75π cm2
6 point 7 5 pi cm squared

13.5π cm2

Answer 1

To find the surface area of the new cylinder after dilation with a scale factor of \( \frac{1}{2} \), we can use the property that when a three-dimensional shape is scaled by a factor \( k \), its surface area scales by a factor of \( k^2 \).

Given:
- Original surface area = \( 54\pi \, \text{cm}^2 \)
- Scale factor (\( k \)) = \( \frac{1}{2} \)

The new surface area is calculated as follows:

\[
\text{New Surface Area} = \text{Original Surface Area} \times k^2
\]

Calculating \( k^2 \):

\[
k^2 = \left( \frac{1}{2} \right)^2 = \frac{1}{4}
\]

Now calculate the new surface area:

\[
\text{New Surface Area} = 54\pi \times \frac{1}{4} = \frac{54\pi}{4} = 13.5\pi \, \text{cm}^2
\]

Thus, the surface area of the new cylinder is:

\[
\boxed{13.5\pi \text{ cm}^2}
\]