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Find the area of the blue shaded region in this 14 by 7 rectangle, with two semicircles of radius 7 drawn.

 

 Dec 30, 2019
 #1
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See the following image :

 

 

 

If we draw  a  line segment  between the two intersection points  of the semi-circles....the length of  this segment - AC -  is given by

 

2√  [ 7^2  - 3.5^2]  =  (2 * 3.5)√ [ 4 - 1] =  7√3

 

Using the Law of Cosines

 

(7√3)^2   = 7^2  + 7^2  -  2 (49) cos (ABC)

 

147  =  98  - 2(49)cos (ABC)

 

49  =  -2(49) cos(ABC)

 

-1/2  =  cos (ABC)

 

arccos (-1/2)  =  ABC  = 

 

ABC  =  120°  =  (2/3)pi

 

So  (1/2)  of the blue shaded area  =

 

Area of sector ABC - area of triangle ABC

 

(1/2) (7^2)(2/3)pi - (1/2)(7^2) sin (120°)

 

(1/2)(7^2)(2/3)pi  -  (1/2)(7^2)*√3/2

 

(1/2) (7^2) [ (2/3) pi  - √3/2 ]

 

So....twice this area =  the blue shaded area  =

 

49  [ (2/3)pi - √3/2 ]  units^2 

 

 

cool cool cool

 Dec 30, 2019
edited by CPhill  Dec 30, 2019

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