The radius of a circle with an area of 60 square centimeters is represented by the expression [tex]$\sqrt{\frac{60}{\pi}}$[/tex] centimeters. What is another way of expressing the radius?

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

Start with the given expression for the radius: π 60 ​ ​ .
Rationalize the denominator by multiplying by π ​ π ​ ​ : π 60 π ​ ​ .
Simplify the radical: 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 the expression π 60 ​ ​ centimeters. We want to find an equivalent expression for this radius from the given options.

Rationalize the denominator To simplify the expression π 60 ​ ​ , we can rationalize the denominator. This means we want to get rid of the π in the denominator inside the square root. To do this, we multiply the fraction inside the square root by π π ​ : π 60 ​ ​ = π 60 ​ ⋅ π π ​ ​ = π 2 60 π ​ ​ Now we can take the square root of the denominator: π 2 60 π ​ ​ = π 2 ​ 60 π ​ ​ = π 60 π ​ ​

Simplify the radical Next, we simplify the radical 60 π ​ . We look for perfect square factors of 60. Since 60 = 4 ⋅ 15 , we have 60 π ​ = 4 ⋅ 15 π ​ = 4 ​ ⋅ 15 π ​ = 2 15 π ​

Substitute back into the expression Substitute this back into the expression: π 60 π ​ ​ = π 2 15 π ​ ​

Final Answer Therefore, the radius can also be expressed as π 2 15 π ​ ​ centimeters.


Examples
The formula for the radius of a circle, r = π A ​ ​ , where A is the area, is used in various real-world applications. For example, if you're designing a circular garden bed and you know the area you want it to cover, you can use this formula to calculate the radius needed. Similarly, in architecture or engineering, if you need a circular component with a specific area, you can determine its radius using this formula. This ensures precise measurements and efficient use of space and materials.

Answered by GinnyAnswer | 2025-07-03

The radius of the circle can be expressed as π 2 15 π ​ ​ centimeters. This is done by rationalizing the denominator of the original expression π 60 ​ ​ . The correct multiple-choice answer is C.
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Answered by Anonymous | 2025-07-04