Question

A cylinder has a surface area of 42 π
cm2 . If it were dilated using a scale factor of 1/2
, what is the surface area of the new cylinder?(1 point)
Responses

21π cm2
21π cm2

13.5π cm2
13.5π cm2

84π cm2
84π cm2

10.5

Answer 1

To find the surface area of the new cylinder after dilating it with a scale factor of \( \frac{1}{2} \), we need to understand how surface area changes with dilation.

When an object is dilated by a scale factor \( k \), the surface area changes by a factor of \( k^2 \). In this case, the scale factor is \( \frac{1}{2} \).

1. Calculate the factor by which the surface area changes:
\[
k^2 = \left( \frac{1}{2} \right)^2 = \frac{1}{4}
\]

2. Now, take the original surface area and multiply it by this factor:
\[
\text{New Surface Area} = \text{Original Surface Area} \times \frac{1}{4}
\]
\[
\text{New Surface Area} = 42 \pi \, \text{cm}^2 \times \frac{1}{4} = \frac{42 \pi}{4} \, \text{cm}^2 = 10.5 \pi \, \text{cm}^2
\]

Therefore, the surface area of the new cylinder is:

\[
\boxed{10.5 \pi \, \text{cm}^2}
\]