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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 a perimeter of 28 cm and an area of 49 sq cm. What is the length of the arc of this sector?

 Oct 24, 2019
 #1
avatar+11646 
+1

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 a perimeter of 28 cm and an area of 49 sq cm. What is the length of the arc of this sector?

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 Oct 24, 2019
 #2
avatar+109064 
+1

Area of sector   =   (1/2)r^2 * theta

Perimeter of sector   =  2r + r *theta =   r [ 2 + theta]

Where theta is the measure of the sector's central angle in radians

 

So

 

49  =  (1/2)r^2 theta   ⇒  49 / [(1/2) r^2]   =  theta     (1)

28  =  r [ 2 + theta]      (2)

 

Subbing (1)  into (2)   we have that

 

28  = r [ 2 +  49 / [(1/2) r^2] ]       simplify

 

28 =  2r + 98/r

 

14  = r + 49/r     multiply through by r

 

14r  = r^2 + 49

 

r^2  - 14r + 49  = 0

 

(r - 7)^2  =  0

 

r  - 7  =  0

 

r  = 7

 

So....the arc length of the sector    =

 

28 - 2(7)  =

 

28  - 14  =

 

14 cm

 

cool cool cool

 Oct 24, 2019

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