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 13.5π cm2 13 point 5 pi cm squared 6.75π cm2 6 point 7 5 pi cm squared 108π cm2
Answers
GPT-4o mini
To determine the surface area of the new cylinder after dilation with a scale factor of 12, we need to understand how surface area changes with scaling.
When a three-dimensional shape is dilated by a scale factor \( k \), the new surface area \( A' \) is given by the formula:
\[
A' = k^2 \times A
\]
Where:
- \( A \) is the original surface area.
- \( k \) is the scale factor.
In this case, the original surface area \( A \) is \( 54\pi \) cm\(^2\), and the scale factor \( k \) is \( 12 \).
So we can calculate the new surface area as follows:
\[
A' = 12^2 \times 54\pi = 144 \times 54\pi
\]
Now calculating \( 144 \times 54 \):
\[
144 \times 54 = 7776
\]
Thus,
\[
A' = 7776\pi \text{ cm}^2
\]
This value is not one of the options provided, likely indicating there was a misunderstanding, perhaps in what the question was asking.
To reevaluate, if we're only interested in direct multiplication:
The scale factor squared \( 12^2 = 144 \).
Then:
\[
A' = 144 \times (54\pi) = 7776\pi
\]
Since this value is significantly larger than those listed, I can only assume that the problem's options might be misperceived because \( A' = 7776\pi \) is indeed the accurate answer.
For clarity sake, let’s double check the interaction with proportions:
To come back to surface area options listed:
- 27π cm²
- 13.5π cm²
- 6.75π cm²
- 108π cm²
These options are incongruent with our scale factor.
In conclusion, assuming fully correct units were mixed, 7776π cm² is the expected output after dilation with the scale factor derived from the formulas above.
When a three-dimensional shape is dilated by a scale factor \( k \), the new surface area \( A' \) is given by the formula:
\[
A' = k^2 \times A
\]
Where:
- \( A \) is the original surface area.
- \( k \) is the scale factor.
In this case, the original surface area \( A \) is \( 54\pi \) cm\(^2\), and the scale factor \( k \) is \( 12 \).
So we can calculate the new surface area as follows:
\[
A' = 12^2 \times 54\pi = 144 \times 54\pi
\]
Now calculating \( 144 \times 54 \):
\[
144 \times 54 = 7776
\]
Thus,
\[
A' = 7776\pi \text{ cm}^2
\]
This value is not one of the options provided, likely indicating there was a misunderstanding, perhaps in what the question was asking.
To reevaluate, if we're only interested in direct multiplication:
The scale factor squared \( 12^2 = 144 \).
Then:
\[
A' = 144 \times (54\pi) = 7776\pi
\]
Since this value is significantly larger than those listed, I can only assume that the problem's options might be misperceived because \( A' = 7776\pi \) is indeed the accurate answer.
For clarity sake, let’s double check the interaction with proportions:
To come back to surface area options listed:
- 27π cm²
- 13.5π cm²
- 6.75π cm²
- 108π cm²
These options are incongruent with our scale factor.
In conclusion, assuming fully correct units were mixed, 7776π cm² is the expected output after dilation with the scale factor derived from the formulas above.
So the answer would be?
GPT-4o mini
The answer would be \( 7776\pi \, \text{cm}^2 \) after dilating the surface area of the cylinder with a scale factor of 12.
However, since \( 7776\pi \, \text{cm}^2 \) is not one of the options you provided, it seems there was a misunderstanding.
To clarify, with respect to the options given (27π, 13.5π, 6.75π, and 108π cm²), **none of those options are correct based on the original surface area of 54π cm² and the dilation factor of 12**.
The correct surface area after dilation with the scale factor of 12 is indeed \( 7776\pi \, \text{cm}^2 \).
However, since \( 7776\pi \, \text{cm}^2 \) is not one of the options you provided, it seems there was a misunderstanding.
To clarify, with respect to the options given (27π, 13.5π, 6.75π, and 108π cm²), **none of those options are correct based on the original surface area of 54π cm² and the dilation factor of 12**.
The correct surface area after dilation with the scale factor of 12 is indeed \( 7776\pi \, \text{cm}^2 \).