Question

A cylinder has a surface area of 42π
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

10.5π cm2
10.5π cm2

Answer 1

When a three-dimensional object, such as a cylinder, is dilated using a scale factor \( k \), the surface area of the new object will be scaled by a factor of \( k^2 \).

In this case, the original surface area of the cylinder is \( 42\pi \) cm², and the scale factor is \( \frac{1}{2} \).

To find the new surface area after dilation, we calculate:

\[
\text{New Surface Area} = \text{Original Surface Area} \times k^2 = 42\pi \, \text{cm}^2 \times \left( \frac{1}{2} \right)^2
\]

Calculating \( k^2 \):

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

Now substituting that back in:

\[
\text{New Surface Area} = 42\pi \, \text{cm}^2 \times \frac{1}{4} = \frac{42\pi}{4} = 10.5\pi \, \text{cm}^2
\]

Thus, the surface area of the new cylinder is \( 10.5\pi \, \text{cm}^2 \).

The correct answer is:

**10.5π cm²**