Final answer:
The scale factor of the dilation from the original circle with an area of 8pi square centimeters to the dilated image with an area of 32pi square centimeters is 2.
Step-by-step explanation:
The student is asking about the scale factor resulting from the dilation of a circle where the original area is 8pi square centimeters and the dilated image's area is 32pi square centimeters. To find the scale factor, start by recognizing that the area of a circle is given by the formula A = πr². If the area of the original circle is 8pi, we can set up an equation 8π = πr², which simplifies to the original radius squared r² being 8. Since the area scales with the square of the scale factor, we can compare the two areas: 8pi and 32pi. This comparison gives us a ratio of 32pi/8pi = 4, indicating that the area of the dilated circle is four times greater than the area of the original circle.
To find the scale factor for the radii (or diameters), we need to take the square root of the area scale factor, because the radius relates to the square root of the area (since A = πr²). So, the scale factor for the radii is the square root of 4, which is 2. Therefore, the scale factor of the dilation from the original to the dilated circle is 2.