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18 Apr 2019, 00:48
Kudos
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E
Be sure to select an answer first to save it in the Error Log before revealing the correct answer (OA)!
Difficulty:
55%
(hard)
Question Stats:
61% (03:42) correct
39%
(03:15)
wrong
based on 18
sessions
History
Date
Time
Result
Not Attempted Yet
The closed curve in the figure is made up of 9 congruent circular arcs each of length \(\frac{2\pi}{3}\), where each of the centers of the corresponding circles is among the vertices of a regular hexagon of side 2. What is the area enclosed by the curve?
(A) \(2\pi + 6\)
(B) \(2\pi + 4\sqrt{3}\)
(C) \(3\pi + 4\)
(D) \(2\pi +3\sqrt{3}+ 2\)
(E) \(\pi + 6\sqrt{3}\)
Attachment:
2019-04-18_1143.png [ 29.45 KiB | Viewed 2946 times ]
18 Apr 2019, 04:11
Kudos
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IMO E
please apply the formula of area of a segment of a circle and the area of a regular hexagon
we will get 5+6√3
Posted from my mobile device
please apply the formula of area of a segment of a circle and the area of a regular hexagon
we will get 5+6√3
Posted from my mobile device
20 Apr 2019, 07:10
Kudos
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Interior angle of Hexagonal = \(\frac{180}{n}\)(n-2)
n=6
So, Interior Angle = 120
Exterior Angle = 240
Perimeter of One congruent circular arc =
\(\frac{120}{360}\) x 2\(\pi\)r =2\(\pi\)/3
So, r = 1
Area of one RED circular segment = \(\frac{240}{360}\) x \(\pi\)\(r^2\)
=2\(\pi\)/3
Area of Three RED circular segment = 3 x 2\(\pi\)/3 = 2\(\pi\)
Side of Hexagonal = 2r
Area of Hexagonal = 3\(\sqrt{3}\)/2 x \(Side^2\) = 6\(\sqrt{3}\)
Area of one Blue circular segment = \(\frac{120}{360}\) x \(\pi\)\(r^2\)
=\(\pi\)/3
Area of Three Blue circular segment = 3 x \(\pi\)/3 = \(\pi\)
AREA OF DESIRED SEGMENT = 6\(\sqrt{3}\)-\(\pi\)+2\(\pi\) = 6\(\sqrt{3}\)+\(\pi\)
n=6
So, Interior Angle = 120
Exterior Angle = 240
Perimeter of One congruent circular arc =
\(\frac{120}{360}\) x 2\(\pi\)r =2\(\pi\)/3
So, r = 1
Area of one RED circular segment = \(\frac{240}{360}\) x \(\pi\)\(r^2\)
=2\(\pi\)/3
Area of Three RED circular segment = 3 x 2\(\pi\)/3 = 2\(\pi\)
Side of Hexagonal = 2r
Area of Hexagonal = 3\(\sqrt{3}\)/2 x \(Side^2\) = 6\(\sqrt{3}\)
Area of one Blue circular segment = \(\frac{120}{360}\) x \(\pi\)\(r^2\)
=\(\pi\)/3
Area of Three Blue circular segment = 3 x \(\pi\)/3 = \(\pi\)
AREA OF DESIRED SEGMENT = 6\(\sqrt{3}\)-\(\pi\)+2\(\pi\) = 6\(\sqrt{3}\)+\(\pi\)
Attachments
Untitled.png [ 18.45 KiB | Viewed 2567 times ]
