Question
A cylinder has a surface area of 42 π
cm2 . If it were dilated using a scale factor of 12
, 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
cm2 . If it were dilated using a scale factor of 12
, 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
Answers
GPT-4o mini
The surface area \( A \) of a cylinder is given by the formula:
\[
A = 2\pi r(h + r)
\]
where \( r \) is the radius and \( h \) is the height.
When a cylinder is dilated using a scale factor \( k \), its new surface area \( A' \) can be calculated using the following relationship:
\[
A' = k^2 \cdot A
\]
In this case, the original surface area \( A \) is \( 42\pi \, \text{cm}^2 \) and the scale factor \( k \) is \( 12 \).
Now we can calculate the new surface area:
\[
A' = 12^2 \cdot 42\pi
\]
Calculating \( 12^2 \):
\[
12^2 = 144
\]
Now multiply by the original surface area:
\[
A' = 144 \cdot 42\pi = 6048\pi \, \text{cm}^2
\]
Thus, the surface area of the new cylinder after dilation is \( 6048\pi \, \text{cm}^2 \).
However, this does not match the options provided. Let's double-check this step since a scale factor larger than 1 indicates that it would increase considerably. If you may meant a much smaller scale factor, please specify.
Given what you presented, make sure if the scale factor is actually 12 or if you need the results with a scale factor <1. Generally, please reconfirm the question.
\[
A = 2\pi r(h + r)
\]
where \( r \) is the radius and \( h \) is the height.
When a cylinder is dilated using a scale factor \( k \), its new surface area \( A' \) can be calculated using the following relationship:
\[
A' = k^2 \cdot A
\]
In this case, the original surface area \( A \) is \( 42\pi \, \text{cm}^2 \) and the scale factor \( k \) is \( 12 \).
Now we can calculate the new surface area:
\[
A' = 12^2 \cdot 42\pi
\]
Calculating \( 12^2 \):
\[
12^2 = 144
\]
Now multiply by the original surface area:
\[
A' = 144 \cdot 42\pi = 6048\pi \, \text{cm}^2
\]
Thus, the surface area of the new cylinder after dilation is \( 6048\pi \, \text{cm}^2 \).
However, this does not match the options provided. Let's double-check this step since a scale factor larger than 1 indicates that it would increase considerably. If you may meant a much smaller scale factor, please specify.
Given what you presented, make sure if the scale factor is actually 12 or if you need the results with a scale factor <1. Generally, please reconfirm the question.