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?

a. 395,688.8 in.^2
b. 915.9 in.^2
c. 305.3 in.^2
d. 1,831.8 in.^2

Answer 1

When a figure is dilated by a scale factor \( k \), the surface area of the dilated figure is related to the original surface area \( A \) by the formula:

\[
\text{Surface Area of Dilated Figure} = k^2 \times \text{Surface Area of Original Figure}
\]

In this case, the scale factor \( k \) is 6, and the surface area of the dilated figure is 10,990.8 in². We can set up the equation as follows:

\[
10,990.8 = 6^2 \times \text{Surface Area of Original Figure}
\]

Calculating \( 6^2 \):

\[
6^2 = 36
\]

Now we can rewrite the equation:

\[
10,990.8 = 36 \times \text{Surface Area of Original Figure}
\]

Now, we solve for the Surface Area of the Original Figure:

\[
\text{Surface Area of Original Figure} = \frac{10,990.8}{36}
\]

Calculating that gives:

\[
\text{Surface Area of Original Figure} = 305.3 \text{ in}^2
\]

Therefore, the surface area of the original figure is \( 305.3 \text{ in}^2 \), which corresponds to option:

**c. 305.3 in²**