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. 305.3 in.^2
B. 395,668.8 in.^2
C. 1,831.8 in.^2
D. 915.9 in.^2
A. 305.3 in.^2
B. 395,668.8 in.^2
C. 1,831.8 in.^2
D. 915.9 in.^2
Answers
GPT-4o mini
To find the surface area of the original figure before dilation, we need to use the relationship between the surface area of a dilated figure and the scale factor. When a figure is dilated by a scale factor of \( k \), the surface area of the dilated figure is multiplied by \( k^2 \).
In this case, the scale factor \( k = 6 \). Therefore, the surface area of the dilated figure (10,990.8 in²) can be expressed in terms of the original surface area \( A \) as follows:
\[
\text{Surface Area of Dilated Figure} = k^2 \times \text{Surface Area of Original Figure}
\]
Substituting in the known values:
\[
10,990.8 = 6^2 \times A
\]
Calculating \( 6^2 \):
\[
6^2 = 36
\]
Now substituting this back in:
\[
10,990.8 = 36 \times A
\]
To find \( A \), divide both sides by 36:
\[
A = \frac{10,990.8}{36}
\]
Calculating \( A \):
\[
A = 305.3 \text{ in}^2
\]
Thus, the surface area of the original figure is:
\(\boxed{305.3 \text{ in}^2}\) (Option A)
In this case, the scale factor \( k = 6 \). Therefore, the surface area of the dilated figure (10,990.8 in²) can be expressed in terms of the original surface area \( A \) as follows:
\[
\text{Surface Area of Dilated Figure} = k^2 \times \text{Surface Area of Original Figure}
\]
Substituting in the known values:
\[
10,990.8 = 6^2 \times A
\]
Calculating \( 6^2 \):
\[
6^2 = 36
\]
Now substituting this back in:
\[
10,990.8 = 36 \times A
\]
To find \( A \), divide both sides by 36:
\[
A = \frac{10,990.8}{36}
\]
Calculating \( A \):
\[
A = 305.3 \text{ in}^2
\]
Thus, the surface area of the original figure is:
\(\boxed{305.3 \text{ in}^2}\) (Option A)