To find the new surface area of a triangular prism after dilation by a scale factor, we need to remember how surface area changes with scale factors.
When an object is dilated using a scale factor of \( k \), the new surface area is calculated by multiplying the original surface area by \( k^2 \) (since surface area is a two-dimensional measurement).
Given:
- Original surface area = 486 in²
- Scale factor \( k = \frac{1}{3} \)
We calculate the new surface area:
\[ \text{New Surface Area} = \text{Original Surface Area} \times k^2 \]
Calculating \( k^2 \):
\[ k^2 = \left(\frac{1}{3}\right)^2 = \frac{1}{9} \]
Now, we calculate the new surface area:
\[ \text{New Surface Area} = 486 \times \frac{1}{9} \]
\[ \text{New Surface Area} = \frac{486}{9} = 54 \text{ in}^2 \]
Thus, the new surface area after dilation with a scale factor of \( \frac{1}{3} \) would be 54 in².
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