When a shape is dilated by a scale factor \( k \), the area of the shape scales by the square of that scale factor, \( k^2 \).
In this case, the scale factor is \( 13 \). So, the new area will be calculated as follows:
\[ \text{New Surface Area} = \text{Original Surface Area} \times k^2 \]
Where:
- Original Surface Area = \( 486 , \text{in}^2 \)
- Scale factor \( k = 13 \)
Calculating \( k^2 \):
\[ k^2 = 13^2 = 169 \]
Now, calculating the new surface area:
\[ \text{New Surface Area} = 486 \times 169 \]
Now let's perform the multiplication:
\[ 486 \times 169 = 82,014 \]
Thus, the new surface area will be:
\[ \text{New Surface Area} = 82,014 , \text{in}^2 \]
It seems like 82,014 in.² is not one of the provided responses. Let's verify if the calculations have any minor errors or possible misunderstanding from the options.
Given the significant dilation factor, usually such a large result won't match the smaller options you provided unless we missed a description. However, the accurate method shows:
The new surface area after applying the dilation is 82,014 in.² and might require a double-check on the selection options provided.