To understand which area equation has been rearranged correctly, let's start with the formula for the area \( A \) of a circle:
\[ A = \pi r^2 \]
If we want to rearrange this equation to solve for radius \( r \), we can do the following steps:
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Divide both sides by \( \pi \): \[ \frac{A}{\pi} = r^2 \]
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Take the square root of both sides to isolate \( r \): \[ r = \sqrt{\frac{A}{\pi}} \]
From the options provided, the correct rearrangement for \( r \) is: \[ r = \sqrt{\frac{A}{\pi}} \] This matches the form of the option \( r = \left( \frac{A}{\pi} \right)^{1/2} \) or equivalently \( r = \frac{A\pi^{-1}}{2} \).
However, that specific option (with the exact notation) seems not presented precisely in your list. The closest to this concept is \( r = \sqrt{A/\pi} \).
Based on what we have discussed, it seems that none of the options provided perfectly reflect the correct rearrangement.