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
27π cm2
27 pi cm squared
108π cm2
108 pi cm squared
6.75π cm2
6 point 7 5 pi cm squared
13.5π cm2
. If it were dilated using a scale factor of 12
, what is the surface area of the new cylinder?(1 point)
Responses
27π cm2
27 pi cm squared
108π cm2
108 pi cm squared
6.75π cm2
6 point 7 5 pi cm squared
13.5π cm2
Answers
GPT-4o mini
The surface area of a cylinder is given by the formula:
\[
\text{Surface Area} = 2\pi r(h + r)
\]
where \( r \) is the radius and \( h \) is the height. When you dilate a shape using a scale factor \( k \), the surface area changes by the scale factor squared (\( k^2 \)).
In this case, the original surface area of the cylinder is \( 54\pi \, \text{cm}^2 \) and the scale factor \( k = 12 \).
To find the surface area of the new cylinder after dilation, you multiply the original surface area by the square of the scale factor:
\[
\text{New Surface Area} = \text{Original Surface Area} \times k^2 = 54\pi \times (12^2)
\]
Calculating \( 12^2 \):
\[
12^2 = 144
\]
Now, multiply the original surface area by 144:
\[
\text{New Surface Area} = 54\pi \times 144 = 7776\pi \, \text{cm}^2
\]
Thus, the surface area of the new cylinder, after dilation, is:
\[
7776\pi \, \text{cm}^2
\]
It appears that my calculation leads to a different answer than the options you provided. Please check if there are any constraints or additional details that may affect the options listed. From your question, the correct surface area of the new cylinder based on the dilation factor provided is indeed \( 7776\pi \, \text{cm}^2 \).
\[
\text{Surface Area} = 2\pi r(h + r)
\]
where \( r \) is the radius and \( h \) is the height. When you dilate a shape using a scale factor \( k \), the surface area changes by the scale factor squared (\( k^2 \)).
In this case, the original surface area of the cylinder is \( 54\pi \, \text{cm}^2 \) and the scale factor \( k = 12 \).
To find the surface area of the new cylinder after dilation, you multiply the original surface area by the square of the scale factor:
\[
\text{New Surface Area} = \text{Original Surface Area} \times k^2 = 54\pi \times (12^2)
\]
Calculating \( 12^2 \):
\[
12^2 = 144
\]
Now, multiply the original surface area by 144:
\[
\text{New Surface Area} = 54\pi \times 144 = 7776\pi \, \text{cm}^2
\]
Thus, the surface area of the new cylinder, after dilation, is:
\[
7776\pi \, \text{cm}^2
\]
It appears that my calculation leads to a different answer than the options you provided. Please check if there are any constraints or additional details that may affect the options listed. From your question, the correct surface area of the new cylinder based on the dilation factor provided is indeed \( 7776\pi \, \text{cm}^2 \).