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 13.5π cm2 13 point 5 pi cm squared 27π cm2
Answers
GPT-4o mini
To determine the surface area of the new cylinder after dilation with a scale factor of 12, we can use the formula for the surface area of a cylinder and the properties of dilations.
1. **Original Surface Area**: The initial surface area of the cylinder is given as \(54\pi \text{ cm}^2\).
2. **Scale Factor**: When an object is dilated by a scale factor \(k\), the surface area changes by the square of the scale factor. In this case, the scale factor is \(12\).
3. **New Surface Area Calculation**: We calculate the new surface area \(A'\) using the formula:
\[
A' = A \cdot k^2
\]
where \(A\) is the original surface area and \(k\) is the scale factor.
Plugging in the values:
\[
A' = (54\pi) \cdot (12^2)
\]
\[
A' = (54\pi) \cdot 144
\]
\[
A' = 7776\pi \text{ cm}^2
\]
Thus, the surface area of the new cylinder after dilation is \(7776\pi \text{ cm}^2\).
However, if you are looking for the possible answers given in the original problem, it seems there is a confusion. Let's just analyze what is given.
The surface area should be significantly larger than any of the options provided (i.e., each of the final options doesn't match this scaling). Please confirm the options if \(7776π\) seems unexpected; otherwise, it is 7776.
If you need surface area uniformly classified in terms of the original radius and surface area properties, confirm for presence of new contexts matching types or geometry interpretations.
1. **Original Surface Area**: The initial surface area of the cylinder is given as \(54\pi \text{ cm}^2\).
2. **Scale Factor**: When an object is dilated by a scale factor \(k\), the surface area changes by the square of the scale factor. In this case, the scale factor is \(12\).
3. **New Surface Area Calculation**: We calculate the new surface area \(A'\) using the formula:
\[
A' = A \cdot k^2
\]
where \(A\) is the original surface area and \(k\) is the scale factor.
Plugging in the values:
\[
A' = (54\pi) \cdot (12^2)
\]
\[
A' = (54\pi) \cdot 144
\]
\[
A' = 7776\pi \text{ cm}^2
\]
Thus, the surface area of the new cylinder after dilation is \(7776\pi \text{ cm}^2\).
However, if you are looking for the possible answers given in the original problem, it seems there is a confusion. Let's just analyze what is given.
The surface area should be significantly larger than any of the options provided (i.e., each of the final options doesn't match this scaling). Please confirm the options if \(7776π\) seems unexpected; otherwise, it is 7776.
If you need surface area uniformly classified in terms of the original radius and surface area properties, confirm for presence of new contexts matching types or geometry interpretations.