Question

cphs : advanced algebra: concepts and connections - block (27.0831030)
dividing radicals
the radius of a circle with an area of 60 square centimeters is represented by the expression \\( \sqrt{\frac{60}{\pi}} \\) centimeters. what is another way of expressing the radius?
options:
\\( \frac{2\sqrt{15\pi}}{\pi} \\)
\\( \frac{4\sqrt{5\pi}}{\pi} \\)
\\( 4\sqrt{5\pi} \\)
\\( 2\sqrt{15\pi} \\)

Explanation:

Step1: Recall the area formula of a circle

The area of a circle is given by \( A=\pi r^{2} \), so we can solve for \( r \): \( r = \sqrt{\frac{A}{\pi}} \). Here, \( A = 60 \), so \( r=\sqrt{\frac{60}{\pi}} \).

Step2: Simplify the radical expression

We can rewrite \( \sqrt{\frac{60}{\pi}} \) as \( \frac{\sqrt{60}}{\sqrt{\pi}} \). Then, simplify \( \sqrt{60} \): \( \sqrt{60}=\sqrt{4\times15} = 2\sqrt{15} \). So now we have \( \frac{2\sqrt{15}}{\sqrt{\pi}} \).

Step3: Rationalize the denominator

To rationalize the denominator, we multiply the numerator and denominator by \( \sqrt{\pi} \): \( \frac{2\sqrt{15}\times\sqrt{\pi}}{\sqrt{\pi}\times\sqrt{\pi}}=\frac{2\sqrt{15\pi}}{\pi} \) (since \( \sqrt{a}\times\sqrt{b}=\sqrt{ab} \) and \( \sqrt{\pi}\times\sqrt{\pi}=\pi \)).

Answer:

\( \boldsymbol{\frac{2\sqrt{15\pi}}{\pi}} \) (which corresponds to the option with this expression, likely the first or fourth option depending on the layout, but based on the simplification, this is the correct form)