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Four semicircular arcs of length 2π are joined to make the figure abov

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Four semicircular arcs of length 2π are joined to make the figure abov  [#permalink]

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New post 30 Aug 2018, 05:11
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79% (02:16) correct 21% (01:24) wrong based on 28 sessions

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Four semicircular arcs of length 2π are joined to make the figure above. What is the area of the enclosed region?


A. \(8\pi\)

B. \(8 + 8\pi\)

C. \(16 + 8\pi\)

D. \(16 + 16\pi\)

E. \(24\pi\)


Attachment:
image019.jpg
image019.jpg [ 2.6 KiB | Viewed 273 times ]

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Re: Four semicircular arcs of length 2π are joined to make the figure abov  [#permalink]

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New post 30 Aug 2018, 06:08
Bunuel wrote:
Image
Four semicircular arcs of length 2π are joined to make the figure above. What is the area of the enclosed region?


A. \(8\pi\)

B. \(8 + 8\pi\)

C. \(16 + 8\pi\)

D. \(16 + 16\pi\)

E. \(24\pi\)


Area of enclosed region=4*Area of semicircle+Area of square (refer affixed diagram)
Given, length of semicircular arc=2π
Or, \(\frac{2πr}{2}=2π\)
Or, r=2

Area of semi-circle=\(\frac{1}{2}*π*r^2=2π\)
Area of square=4*(2r)=16

So , area of enclosed region=4*2π+16=8π+16

Ans. (C)
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Re: Four semicircular arcs of length 2π are joined to make the figure abov  [#permalink]

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New post 30 Aug 2018, 06:08
Top Contributor
Bunuel wrote:
Image
Four semicircular arcs of length 2π are joined to make the figure above. What is the area of the enclosed region?


A. \(8\pi\)

B. \(8 + 8\pi\)

C. \(16 + 8\pi\)

D. \(16 + 16\pi\)

E. \(24\pi\)

Attachment:
image019.jpg


If one SEMI-circle has a perimeter of 2π, then an ENTIRE circle must have a circumference of
Image

Circumference = (diameter)(π)
So, if the circumference of the circle is , then the diameter of each circle is 4
Image

This means the square in the middle of the figure must have area 16.
Image

Now we need only find the area of 4 SEMIcircles.
If we combine 2 semicircles we get an entire circle.
So, if we combine 4 semicircles we get 2 entire circles.

Each circle has diameter 4, which means each circle has RADIUS 2
Area of a circle = π(radius²)
So, area of one circle = π(2²) = 4π
So, the area of TWO circles = 8π

So, the TOTAL area = 16 + 8π

Answer: C

Cheers,
Brent
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Re: Four semicircular arcs of length 2π are joined to make the figure abov &nbs [#permalink] 30 Aug 2018, 06:08
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