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

84π cm2
84π cm2

21π cm2
21π cm2

13.5π cm2
13.5π cm2

10.5π cm2
10.5π cm2

Answer 1

When a three-dimensional shape like a cylinder is dilated by a scale factor \(k\), the surface area \(A\) of the shape changes by the square of that scale factor.

In this case, the original surface area \(A = 42\pi \, \text{cm}^2\) and the scale factor \(k = \frac{1}{2}\).

The new surface area \(A'\) can be calculated as follows:

\[
A' = A \times k^2
\]

Substituting the values:

\[
A' = 42\pi \times \left(\frac{1}{2}\right)^2
\]
\[
A' = 42\pi \times \frac{1}{4}
\]
\[
A' = \frac{42\pi}{4} = 10.5\pi \, \text{cm}^2
\]

So, the surface area of the new cylinder is:

**10.5π cm²**.