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Two semicircles are drawn in a 14 by 7 rectangle.  Find the area of the blue region.

 

 Dec 27, 2020
 #1
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NO GUARANTEES:   (YOU should check my math)

 

 Dec 27, 2020
 #2
avatar+283 
+3

To answer this question, you will need to find the area of the chord of a circle with a radius of 7, shown in the image below, and then multiply by 2

The area of the chord is the area of the sector minus the area of the triangle:

\(\frac{49\pi}{3}-3.5\sqrt{3}\cdot3.5\cdot\frac{1}{2}\cdot2 \)

=\(\frac{49\pi}{3}-12.25\sqrt{3}\)

 

Now, multiply that by 2 and the answer is \(\frac{98\pi}{3}-25\sqrt{3}\)

That is roughly equal to 59.324

 Dec 27, 2020
 #3
avatar+117576 
+1

See the following :

 

We  have   30-60-90  triangle GIF   where  GI  = 3.5  and  IF =  3.5sqrt (3)

So....the  area of this triangle is  (1/2) (3.5)(3/5) sqrt (3)  = (49/4)sqrt (3)/2            (1)

 

And then we have   the  area  of   60°  sector   of a circle with a rdius of 7  less a triangle with sides of  7  and an included angle of 60°

 

So this area is    (1/2)7^2 ( pi/3)  - (1/2) ( 7)^2 sqrt (3) / 2  =  (49/2)  ( pi/3 - sqrt (3) / 2)   =

49pi/6 - (49/4) sqrt (3)     (2)

 

So   1/4 area  of this  shaded region  is  (1) + (2)   =    (49/6) pi - (49sqrt (3)  /8)

 

So....the total area is    4  [ (49/6) pi  - 49sqrt (3) / 8 '  =

 

(98/3) pi  - (49/2) sqrt (3)    ≈  60.19  units^2

 

Exactly  what Guest found   !!!

 

cool cool cool

 Dec 27, 2020
edited by CPhill  Dec 27, 2020

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