To find the scale factor of the dilation, we need to understand the relationship between the area of the original figure and the area of the dilated figure.
When a two-dimensional shape is dilated, the area is multiplied by the square of the scale factor. That is, if the scale factor is \( k \), and a shape's original area \( A \) is scaled, the new area \( A' \) is given by:
\[ A' = k^2 \cdot A \]
In this particular case, we have the original area of the circle as \( 8x \) square centimeters, and the dilated circle's area as \( 32x \) square centimeters. Therefore, according to the area scale factor relationship, we have:
\[ 32x = k^2 \cdot 8x \]
Now, to find \( k \) (the scale factor), we need to isolate \( k^2 \) by dividing both sides of the equation by the original area:
\[ k^2 = \frac{32x}{8x} \]
Since \( x \) is present on both sides of the equation, it simplifies out:
\[ k^2 = \frac{32}{8} \]
\[ k^2 = 4 \]
Finally, we take the square root of both sides of the equation to solve for \( k \):
\[ k = \sqrt{4} \]
\[ k = 2 \]
Therefore, the scale factor of the dilation is \( 2 \).
