To find the surface area of the dilated cylinder, we need to remember how surface area changes with dilation.
When an object is dilated by a scale factor \( k \), the new surface area \( A' \) can be calculated using the formula:
\[ A' = k^2 \times A \]
where \( A \) is the original surface area.
In this case, the original surface area of the cylinder is \( 54\pi , \text{cm}^2 \) and the scale factor \( k \) is \( 12 \).
Now we calculate:
\[ A' = (12^2) \times (54\pi) = 144 \times (54\pi) \]
Calculating \( 144 \times 54 \):
\[ 144 \times 54 = 7776 \]
Therefore,
\[ A' = 7776\pi , \text{cm}^2 \]
Now we check to see if this matches any of the provided responses:
- \( 108\pi , \text{cm}^2 \)
- \( 6.75\pi , \text{cm}^2 \)
- \( 13.5\pi , \text{cm}^2 \)
- \( 27\pi , \text{cm}^2 \)
None of the provided options match \( 7776\pi , \text{cm}^2 \).
I realize I must have made a mistake while multiplying. Let's go back step-by-step.
Calculating \( 12^2 \):
\[ 12^2 = 144 \]
Now we calculate:
\[ 144 \times 54 = 7776 \]
The original surface area \( A \) is \( 54\pi \).
So we need \( 144 \cdot 54 \):
Let's redo that calculation one more time:
\[ 144 \times 54 = (140 + 4) \times 54 = 140 \times 54 + 4 \times 54 = 7560 + 216 = 7776 \]
It seems the provided responses don't show \( 7776\pi \) as an option, indicating that an assumption or option is incorrect.
Rechecking our options and calculations leads us to confirm \( A' \) is indeed \( 7776\pi , \text{cm}^2 \), but no matching options are listed. Please verify option correctness or any constraints you might have.
In standard dilation calculations based on surface area, \( k^2 \) is the correct transformation based on original forms, leading to \( 7776\pi \). Conclusions are based on fundamental geometric principles.
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