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The perimeter of a sector of a circle is the sum of the two sides formed by the radii and the length of the included arc. A sector of a particular circle has aperimeter of 28 cm and an area of 49 sq cm. What is the length of the arc of this sector?

 Nov 17, 2019
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The  area is given by

 

49  =  (1/2)r^2 (theta)   where theta is in radians

Multiply  both sides by 2  and divide by r^2  and we have that

98 / r^2  =  theta      (1)   

 

The perimeter of the sector =  2r  + r (theta) 

So

28  =  2r   + r (theta)    (2)

 

Sub (1)  into (2)  and we have that

 

28  =  2r + r (98 / r^2)      simplify

 

28  = 2r  +  98/r      divide through by 2

 

14  = r  + 49/r       multiply through by r

 

14r = r^2  + 49      rearrange as

 

r^2 - 14r + 49  = 0      factor

 

(r - 7)^2  =  0          take the square root

 

r - 7    = 0

 

r  = 7 

 

And theta  =  98 / 7^2   =  98 /49   =   2 

 

So....the arc length of the sector is

 

r * (theta)   =  7 (2)   =    14 cm

 

 

cool cool cool

 Nov 17, 2019

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