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shaded semicircles

by sanju09 » Wed Mar 04, 2009 4:55 am
In the figure attached, the circle has center Y and the four shaded semicircles each have area 18 pi. What is the circumference of the circle?

(A) 24 pi
(B) 30 pi
(C) 36 pi
(D) 48 pi
(E) 60 pi
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Last edited by sanju09 on Wed Mar 04, 2009 5:08 am, edited 1 time in total.
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by truplayer256 » Wed Mar 04, 2009 5:01 am
Area of one semi-circle=18pi
pi*r^2/2=18pi
pi*r^2=36pi
r^2=36
r=6
2r=d
The diameter of one semicircle is 12. Since there are 4 semicircles, the diameter of the circle is 12*4=48.
C=Pi*d
C=pi*48=48pi D
Last edited by truplayer256 on Wed Mar 04, 2009 5:14 am, edited 1 time in total.
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by sanju09 » Wed Mar 04, 2009 5:09 am
truplayer256 wrote:Area of one semi-circle=18
pi*r^2/2=18
pi*r^2=36
r^2=36/pi
r=6/sqrt(pi)
2r=d
The diameter of one semicircle is 12/sqrt(pi). Since there are 4 semicircles, the diameter of the circle is 12/sqrt(pi)*4=48/sqrt(pi)
C=Pi*d
C=pi*48/sqrt(pi)=48sqrt(pi).. not an answer choice?
:) Try again with "the four shaded semicircles each have area 18 pi" now.
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Re: shaded semicircles

by farooq » Wed Mar 04, 2009 5:15 am
sanju09 wrote:In the figure attached, the circle has center Y and the four shaded semicircles each have area 18 pi. What is the circumference of the circle?

(A) 24 pi
(B) 30 pi
(C) 36 pi
(D) 48 pi
(E) 60 pi
Answer is D.

semi circle area = 18 pi. So the full area will be 36 pi and its radius=6, diameter = 12.

therefore radius of the bigger circle comes 24 and its circum=2*24*pi = 48pi
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Re: shaded semicircles

by marcusking » Wed Mar 04, 2009 10:05 am
sanju09 wrote:In the figure attached, the circle has center Y and the four shaded semicircles each have area 18 pi. What is the circumference of the circle?

(A) 24 pi
(B) 30 pi
(C) 36 pi
(D) 48 pi
(E) 60 pi
If semi circle is 18pi full circle must be 36pi meaning r of semi circle must be 6 so each semi circles diameter is 12.

There are 4 semi circles that span the diameter of the larger circle
4 * 12 = 48

Circumference = pi(diameter)
Circumference = 48pi
D.

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by deepoe » Wed Mar 04, 2009 10:16 am
The 36PI is the area of one full area or not?
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by sanju09 » Thu Mar 05, 2009 3:43 am
The circumference of the circle is equal to D pi , where D is the diameter of the circle. The diameter of the circle is the sum of the diameters of the four semicircles. The four semicircles have equal area, so they must also have equal radii. Let r be the radius of one of the semicircles. Each of the four semicircles has area 18 pi, so 18 pi = ½ pi r^2. Solving for r gives r = 6. Therefore, the sum of the diameters of the four semicircles is 2 r + 2 r + 2 r + 2 r = 8 r = 48. Thus 48 is also the diameter D of the circle, and the circumference of the circle is [spoiler]D pi = 48 pi[/spoiler], which is choice [spoiler](D)[/spoiler].
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