To find the new surface area of a dilated figure, you can use the formula that relates the surface area of similar figures:
\[ \text{New Surface Area} = \text{Original Surface Area} \times \text{Scale Factor}^2 \]
In this case, the original surface area of the triangular prism is 486 in² and the scale factor is 13.
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First, calculate the square of the scale factor: \[ 13^2 = 169 \]
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Then, multiply the original surface area by this factor: \[ \text{New Surface Area} = 486 , \text{in}^2 \times 169 \]
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Now, perform the multiplication: \[ \text{New Surface Area} = 486 \times 169 = 82,014 , \text{in}^2 \]
Double check: \[ \text{New Surface Area} = 486 \times 169 = 82,014 \]
So, the new surface area would be 82,014 in².
However, since that answer does not match your options, let’s check the scale factor again: If there had been an error in the scale factor value, for instance, if it were meant to refer to a different number, you would scale the original area using up the equation correctly.
For 3D figures, make sure to always square the linear scale factor to determine the new surface areas correctly.
If you indeed meant to have the scale factor as 13, the answer should have been provided as 82,014 in², provided that all calculations are correct and were based on the originally stated factor of scaling.
Final answer should then be confirmed as: Based on inputs given, please verify the scale factor or the area to ensure all data aligns with problem expectations.
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