To find the surface area of the new cylinder after dilation, we can use the properties of similar shapes. When an object is dilated by a scale factor \( k \), its surface area is affected by a factor of \( k^2 \).
Given:
- Original surface area = \( 54\pi , \text{cm}^2 \)
- Scale factor = \( 12 \)
Now, calculate the scaled surface area:
-
Calculate \( k^2 \): \[ k^2 = 12^2 = 144 \]
-
Calculate the new surface area: \[ \text{New surface area} = \text{Original surface area} \times k^2 = 54\pi \times 144 \] \[ \text{New surface area} = 7776\pi , \text{cm}^2 \]
Now that we have computed the new surface area, it seems that the options provided do not include \( 7776\pi \). Hence, it's likely that I made an error. Let's check this again:
The correct approach is as follows:
- When a cylinder’s dimensions (height and radius) are dilated by a factor of \( 12 \), then its surface area will be multiplied by \( 12^2 = 144 \).
So: \[ \text{New surface area} = 54\pi \times 144 \] \[ = (54 \times 144)\pi , \text{cm}^2 \] \[ = 7776\pi , \text{cm}^2, \]
It appears I initially misinterpreted your options in context. The calculations show that the new surface area based on the dilation is \( 7776\pi , \text{cm}^2 \) as derived from the scale factor. This exceeds the options presented, indicating either an oversight or error in the problem context provided.
None of the provided options match the computed dilation outcome from the surface area calculation. Please recheck the inputs or consider if there might be simplification assumptions that were intended to be accounted for differently!
1 answer