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If a central angle of $90$ degrees defines an arc on circle $R$ that has the same length as the arc on circle $W$ defined by a $60$-degree central angle, what is the ratio of the area of circle $R$ to the area of circle $W$? Express your answer as a common fraction.

 Jun 27, 2018
 #1
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Arc Length = [90 x pi/180 x R] =[60 x pi/180 x W], solve for R/W
R/W = 2/3 - ratio of the area of circle R to circle W

 Jun 27, 2018
 #2
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Let R1  be the radius of the first circle

And let R2 be the measure of the second circle

 

So

 

Arc length  =   radius * theta ( in rads )

 

So....since the arc lenghts are equal....

 

R1 * pi/2   = R2 * pi/3

 

R1 / 2  =  R2 / 3

 

R1  =  (2/3)R2

 

So...the area of  the first circle  is

[ pi * (R1) ^2 ]  =    pi *[  (2/3) R2 ] ^2  =  pi  (4/9) (R2)^2

 

And the area of the second circle  is

pi [ R2]^2  

 

So  the ratio  of the area  of the first cirlce to the second  is

 

[ pi * (4/9) (R2)^2 ]               4

_______________  =       ____

[ pi * (R2)^2 ]                        9

 

 

cool cool cool

 Jun 27, 2018

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