To find the new surface area of a cylinder after a dilation with a scale factor of 4, we need to understand how the surface area of a 3D object scales with its linear dimensions.
When a cylinder (or any 3D shape) is dilated by a scale factor \( k \), the new surface area \( SA' \) can be found using the following relationship:
\[ SA' = k^2 \cdot SA \]
where \( SA \) is the original surface area, and \( k \) is the scale factor.
Let's start by recalling the formula for the surface area of a cylinder:
\[ SA = 2\pi rh + 2\pi r^2 \]
This formula can be interpreted as the lateral surface area \( (2\pi rh) \) plus the area of the two bases \( (2\pi r^2) \).
Now, setting the scale factor:
Given that the scale factor \( k = 4 \), we can square this value:
\[ k^2 = 4^2 = 16 \]
This means the new surface area will be:
\[ SA' = 16 \cdot SA \]
Thus, the new surface area is 16 times the original surface area.
If you have the original surface area \( SA \), you can multiply it by 16 to find the new surface area. However, since the original surface area is not provided, we can only conclude that the new surface area will be 16 times the original value.
In the answer options provided, we cannot directly pick a value without knowing the original surface area. If one of those options represents \( SA \) and multiplying by 16 gives one of the choices, that would be the new surface area.
Please provide the original surface area of the cylinder so we can accurately calculate the new surface area.