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A sector of a circle is shown below.  The sector has an area of $60 \pi.$  What is the radius of the circle?

 

 Dec 17, 2023
 #1
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Knowing the angle of the sector and its area, we can find the radius of the circle as follows:

 

Area of the whole circle: Since the sector covers half the circle (180 degrees), and the circle's area is equal to the area of a full sector with 360 degrees, we can find the entire circle's area by doubling the sector's area:

Circle area = 2 * Sector area = 2 * 60π = 120π

 

Relationship between area and radius: The area of a circle is related to its radius (r) by the formula:

Circle area = π * r^2

 

Solving for the radius: Substitute the calculated whole circle area in the formula:

120π = π * r^2

 

Divide both sides by π:

120 = r^2

 

Take the square root of both sides:

r = 10√2

 

Therefore, the radius of the circle is 10√2 units.

 Dec 17, 2023
 #2
avatar+676 
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A sector of a circle is shown below.  The sector has an area of $60 \pi.$  What is the radius of the circle?  

 

I'm not sure what \pi means.  Does the backslant mean divided by pi, or is the backslant just another one of those useless symbols like all those dollar signs in every problem I look at.  I don't know about you, but I would use a forward slant to mean divided by.  I'm going to treat the backslant as meaningless.  

 

The shaded area is a fourth of the circle, indicated by the right angle in the center. 

 

                                                          Acircle  

                                                        –––––––  =  60 π  

                                                             4  

 

                                                           A  =  240 π  

 

                                                           A  =  π r2   

 

therefore                                       240 π  =  π r2     

 

                                                            r2  =  240  

 

                                                            r  =  sqrt(240)  

 

                                                            r  =  4 • sqrt(15)  

.

 Dec 18, 2023

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