This problem is from the Quant Review
Figure shows 3 identical circles that are tangent to each other. If the area of the shaded region is 64Root3-32PI, what is the radius of each circle?
OG 8
This is how I attacked the problem.
First I noticed that the triangle formed was an equilateral triangle. I know if I find the side of that I can multiply by 1/2 and get the radius of the circle.
I used the formula for the equilateral and take a look at the equation given. The equilateral equation is (S^2ROOT3)/4. The 64 root 3 part looks familiar so I multiply the 64 by 4 and get 256. I know that 16^2 is 256 so r must be 1/2*16 which is 8.
Is this the best way to do this problem?
3 Tangent circles
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Shaded region = triangle - 3 circle sectors = 64√3 - 32�
Looking at the equation above, we can see that the 3 circle sectors = 32�.
Since the triangle is equilateral, each of its angles is 60 degrees. Since 60/360 = 1/6, each circle sector is 1/6 the area of each circle.
Thus, the 3 sectors = 3*1/6 = 1/2 each circle area.
Since the 3 sectors = 32�, circle area = 64�.
Thus, �r² = 64�, and r=8.
The correct answer is B.
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let the radius of a circle be x
since the triangle is equilateral, its angle is equal to 60degree
the bottom of the triangle is 2x, the diagonal is x*sqroot3
the area of the triangle =2x*x*sqroot3/2=(x^2)*sqroot3
the area of the sector of a cirle =(60/360)*pi*x^2=(1/6)*pi*x^2
so we have -
(x^2)*sqroot3 -3* (1/6)*pi*x^2=64sqroot3-32pi
x^2(sqroot3-(1/2)*pi*x^2)=32(2sqroot3-pi)
x^2=32*2
x=8
since the triangle is equilateral, its angle is equal to 60degree
the bottom of the triangle is 2x, the diagonal is x*sqroot3
the area of the triangle =2x*x*sqroot3/2=(x^2)*sqroot3
the area of the sector of a cirle =(60/360)*pi*x^2=(1/6)*pi*x^2
so we have -
(x^2)*sqroot3 -3* (1/6)*pi*x^2=64sqroot3-32pi
x^2(sqroot3-(1/2)*pi*x^2)=32(2sqroot3-pi)
x^2=32*2
x=8
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