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 changes with respect to the scale factor.
The surface area \( SA \) of a cylinder is given by the formula:
\[ SA = 2\pi rh + 2\pi r^2 \]
When a shape is dilated by a scale factor \( k \), the new dimensions of the shape (height and radius) will also be scaled by that factor \( k \). The new radius \( r' \) and height \( h' \) after dilation would be:
\[ r' = k \cdot r = 4r \] \[ h' = k \cdot h = 4h \]
To find the new surface area \( SA' \), we substitute \( r' \) and \( h' \) into the surface area formula:
\[ SA' = 2\pi (r')(h') + 2\pi (r')^2 \] \[ SA' = 2\pi (4r)(4h) + 2\pi (4r)^2 \] \[ SA' = 2\pi (16rh) + 2\pi (16r^2) \] \[ SA' = 16(2\pi rh) + 16(2\pi r^2) \] \[ SA' = 16 (2\pi rh + 2\pi r^2) \] \[ SA' = 16 \cdot SA \]
Thus, the surface area increases by a factor of \( 16 \) when the cylinder is dilated by a scale factor of \( 4 \).
So, if you have the original surface area, you would multiply it by \( 16 \) to find the new surface area. However, the original surface area is not provided in the question.
If you have the original surface area (let's call it \( SA \)), then the new surface area will be:
\[ SA' = 16 \cdot SA \]
If \( SA \) corresponds to a particular option, compute \( 16 \cdot SA \) and compare with the options:
- \( 1,472\pi , cm^2 \)
- \( 5,888\pi , cm^2 \)
- \( 368\pi , cm^2 \)
- \( 23\pi , cm^2 \)
So depending on the original surface area, substitute that value into the \( 16 \cdot SA \) formula and match it to the available options. If you have the original surface area number, please provide it for a more direct answer.