Question

Similarity Unit Test
13 of 2013 of 20 Items
























Question
A cylinder has a surface area of 54π cm2
. If it were dilated using a scale factor of 12
, what is the surface area of the new cylinder?(1 point)
Responses

108π cm2
108 pi cm squared

6.75π cm2
6 point 7 5 pi cm squared

27π cm2
27 pi cm squared

13.5π cm2
13 point 5 pi cm squared
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Answer 1

To determine the surface area of the new cylinder after dilation, we first need to understand how dilation affects surface area.

The formula for the surface area of a cylinder is given by:

\[ \text{Surface Area} = 2\pi r^2 + 2\pi rh \]

When a three-dimensional object is dilated by a scale factor \( k \), the new surface area \( A' \) relates to the original surface area \( A \) as follows:

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

In this case, the original surface area \( A \) is \( 54\pi \, \text{cm}^2 \) and the scale factor \( k \) is \( \frac{1}{2} \).

Now, calculate the new surface area:

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

2. Calculate the new surface area:
\[
A' = \frac{1}{4} \times 54\pi = \frac{54\pi}{4} = 13.5\pi \, \text{cm}^2
\]

Thus, the surface area of the new cylinder after dilation is **13.5π cm²**.

So the correct response from the options is:

**13.5π cm²**