Question

the formula for the area of a circle is given, where a is the area, in square units, and r is the radius of the circle, in units.
$a = \pi r^2$
which equation shows r in terms of a?
options:
\\( r = \sqrt{\frac{a}{\pi}} \\) (assuming the correct option formatting, with other options as similar algebraic rearrangements)

Explanation:

Step1: Start with the area formula

We have the formula for the area of a circle: \( A=\pi r^{2} \). Our goal is to solve for \( r \) in terms of \( A \).

Step2: Isolate \( r^{2} \)

Divide both sides of the equation \( A = \pi r^{2} \) by \( \pi \). This gives us \( \frac{A}{\pi}=r^{2} \).

Step3: Solve for \( r \)

Take the square root of both sides to solve for \( r \). Since the radius \( r \) is a non - negative quantity (as it represents a length), we have \( r=\sqrt{\frac{A}{\pi}} \) (we consider the positive square root because radius cannot be negative).

Answer:

The equation that shows \( r \) in terms of \( A \) is \( r = \sqrt{\frac{A}{\pi}} \) (assuming the second option in the list is \( r=\sqrt{\frac{A}{\pi}} \), the answer would be that option. For example, if the options are labeled as (A) \( r=\sqrt{\pi A} \), (B) \( r = \sqrt{\frac{A}{\pi}} \), (C) \( r=\pi\sqrt{A} \), (D) \( r=\sqrt{\frac{\pi}{A}} \), then the answer is B. \( r=\sqrt{\frac{A}{\pi}} \))