Question

A dilated figure has a surface area of 10,990.8 in.^2. The figure was dilated by a scale factor of 6. What is the surface area of the original figure?

395,668.8 in.^2
1,831.8 in.^2
915.9 in.^2
305.3 in.^2

Answer 1

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:

\[
\boxed{305.3 \, \text{in}^2}
\]