Question
A dilated figure has a surface area of 10,990.8 in.2. The figure dilated by a scale of 6. What is the surface area of the original figure
Answers
GPT-4o mini
When a figure is dilated by a scale factor \( k \), the surface area of the dilated figure is affected by the square of the scale factor. This means that if the original surface area is \( A \) and the scale factor is \( k \), the surface area of the dilated figure becomes:
\[
A_{\text{dilated}} = k^2 \times A
\]
In this case, the scale factor \( k = 6 \), so:
\[
A_{\text{dilated}} = 6^2 \times A = 36 \times A
\]
Given that the surface area of the dilated figure is \( 10,990.8 \, \text{in}^2 \), we can set up the equation:
\[
10,990.8 = 36 \times A
\]
To find the surface area of the original figure \( A \), we need to solve for \( A \):
\[
A = \frac{10,990.8}{36}
\]
Calculating \( A \):
\[
A = 305.3 \, \text{in}^2
\]
Thus, the surface area of the original figure is \( 305.3 \, \text{in}^2 \).
\[
A_{\text{dilated}} = k^2 \times A
\]
In this case, the scale factor \( k = 6 \), so:
\[
A_{\text{dilated}} = 6^2 \times A = 36 \times A
\]
Given that the surface area of the dilated figure is \( 10,990.8 \, \text{in}^2 \), we can set up the equation:
\[
10,990.8 = 36 \times A
\]
To find the surface area of the original figure \( A \), we need to solve for \( A \):
\[
A = \frac{10,990.8}{36}
\]
Calculating \( A \):
\[
A = 305.3 \, \text{in}^2
\]
Thus, the surface area of the original figure is \( 305.3 \, \text{in}^2 \).