Question

what is the area of the image, given a scale factor of (\frac{1}{8}), if the area of the preimage is (576 km^2)? (\bigcirc 72 km^2) (\bigcirc 9 km^2) (\bigcirc 576 km^2) (\bigcirc 4,608 km^2)

Explanation:

Step1: Recall area scale factor rule

When a figure is scaled by a scale factor \( k \), the area of the image is the area of the preimage multiplied by \( k^2 \). Here, the scale factor \( k=\frac{1}{8} \), so the area scale factor is \( k^2 = (\frac{1}{8})^2=\frac{1}{64} \).

Step2: Calculate the area of the image

The area of the preimage is \( 576\space km^2 \). Multiply the preimage area by the area scale factor: \( 576\times\frac{1}{64} \).
Simplify \( 576\div64 = 9 \). So the area of the image is \( 9\space km^2 \).

Answer:

\( 9\space km^2 \) (corresponding to the option "9 \( km^2 \)")