When a shape is dilated by a scale factor \( k \), the surface area of the shape changes by a factor of \( k^2 \).
In this case, the scale factor is \( \frac{1}{3} \).
To find the new surface area:
-
Calculate \( k^2 \): \[ k^2 = \left(\frac{1}{3}\right)^2 = \frac{1}{9} \]
-
Multiply the original surface area by this factor: \[ \text{New Surface Area} = \text{Original Surface Area} \times k^2 = 486 \times \frac{1}{9} \]
-
Now, perform the multiplication: \[ 486 \div 9 = 54 \]
Thus, the new surface area after dilation is \( 54 , \text{in}^2 \).
The correct answer is: O 54 in?