When a three-dimensional shape, such as a triangular prism, is dilated by a scale factor \( k \), the surface area of the shape is affected by the square of that scale factor.
The relationship for the surface area after dilation can be calculated with the following formula:
\[ \text{New Surface Area} = \text{Original Surface Area} \times k^2 \]
Given:
- Original Surface Area = 486 in²
- Scale Factor \( k = \frac{1}{3} \)
Now, calculate \( k^2 \):
\[ k^2 = \left( \frac{1}{3} \right)^2 = \frac{1}{9} \]
Now, substitute the values into the new surface area formula:
\[ \text{New Surface Area} = 486 \times \frac{1}{9} \]
To perform the multiplication:
\[ \text{New Surface Area} = \frac{486}{9} = 54 , \text{in}^2 \]
Thus, the new surface area after the dilation is:
\[ \boxed{54 , \text{in}^2} \]
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