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?

A. $2 \sqrt{15 \pi}$
B. $4 \sqrt{5 \pi}$
C. $\frac{2 \sqrt{15 \pi}}{\pi}$
D. $\frac{4 \sqrt{5 \pi}}{\pi}$

The equivalent expression for the radius of the circle is π 2 15 π ​ ​ centimeters, matching option C. This was derived by rationalizing the denominator and simplifying the radical expression. The final answer clearly shows how to transform the area into an expression for the radius in a different form.
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Answered by Anonymous | 2025-07-04

Start with the expression for the radius: π 60 ​ ​ .
Rationalize the denominator by multiplying by π ​ π ​ ​ , resulting in π 60 π ​ ​ .
Simplify the radical: 60 π ​ = 2 15 π ​ .
The equivalent expression for the radius is π 2 15 π ​ ​ ​ .

Explanation

Problem Analysis The radius of a circle with area 60 square centimeters is given by π 60 ​ ​ centimeters. We want to find an equivalent expression for this radius from the given options.

Rationalize the denominator and simplify the radical We start with the given expression for the radius: π 60 ​ ​ To rationalize the denominator, we multiply the expression by π ​ π ​ ​ : π 60 ​ ​ ⋅ π ​ π ​ ​ = π 60 π ​ ​ Now, we simplify the radical in the numerator. We look for perfect square factors of 60 :
60 = 4 ⋅ 15 , so 60 π ​ = 4 ⋅ 15 π ​ = 4 ​ ⋅ 15 π ​ = 2 15 π ​ .

Substitute back into the expression Substitute the simplified radical back into the expression: π 2 15 π ​ ​

Compare with the given options Now we compare our simplified expression with the given options: 2 15 π ​ 4 5 π ​ π 2 15 π ​ ​ π 4 5 π ​ ​


The expression we found matches the third option.

Final Answer Therefore, another way of expressing the radius is π 2 15 π ​ ​ centimeters.

Examples
Understanding how to simplify radical expressions like this is useful in various fields, such as engineering and physics, where you often deal with formulas involving square roots and fractions. For example, when calculating the impedance of an electrical circuit or the period of a pendulum, you might encounter similar expressions that need simplification to obtain a more usable form. Simplifying these expressions makes calculations easier and helps in understanding the relationships between different variables.

Answered by GinnyAnswer | 2025-07-04