When a figure is dilated by a scale factor \( k \), the surface area of the dilated figure is given by the formula:
\[ \text{Surface Area}{\text{dilated}} = k^2 \times \text{Surface Area}{\text{original}} \]
In this case, the scale factor \( k = 6 \) and the surface area of the dilated figure is \( 10,990.8 , \text{in}^2 \). We can set up the equation:
\[ 10,990.8 = 6^2 \times \text{Surface Area}_{\text{original}} \]
Calculating \( 6^2 \):
\[ 6^2 = 36 \]
Substituting this value into the equation gives:
\[ 10,990.8 = 36 \times \text{Surface Area}_{\text{original}} \]
Now, to find the surface area of the original figure, we can solve for \( \text{Surface Area}_{\text{original}} \):
\[ \text{Surface Area}_{\text{original}} = \frac{10,990.8}{36} \]
Calculating this gives:
\[ \text{Surface Area}_{\text{original}} = 305.3 , \text{in}^2 \]
Thus, the surface area of the original figure is:
\[ \boxed{305.3 , \text{in}^2} \]
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