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# The circle shown above has center O and radius of length 5. If the are

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Math Expert
Joined: 02 Sep 2009
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The circle shown above has center O and radius of length 5. If the are [#permalink]

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16 Nov 2017, 00:46
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The circle shown above has center O and radius of length 5. If the area of the shaded region is 20π, what is the value of x?

(A) 18
(B) 36
(C) 45
(D) 54
(E) 72

[Reveal] Spoiler:
Attachment:

2017-11-16_1236_002.png [ 5.5 KiB | Viewed 539 times ]
[Reveal] Spoiler: OA

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Intern
Joined: 08 Aug 2017
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Re: The circle shown above has center O and radius of length 5. If the are [#permalink]

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16 Nov 2017, 03:25
Area of the entire circle - area of the unshaded region(x) = area of the shaded region....(1)
area of the entire circle = 25pi
area of shaded region = 20pi
pluggin in (1), 25pi - x = 20pi
x = 5pi
5pi/25pi = degree measure/ 360
degree measure = 72
now, in triangle x = 180-90-72 = 18

hence, A.

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VP
Joined: 22 May 2016
Posts: 1131

Kudos [?]: 403 [0], given: 645

The circle shown above has center O and radius of length 5. If the are [#permalink]

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16 Nov 2017, 10:17
Bunuel wrote:

The circle shown above has center O and radius of length 5. If the area of the shaded region is 20π, what is the value of x?

(A) 18
(B) 36
(C) 45
(D) 54
(E) 72

[Reveal] Spoiler:
Attachment:
2017-11-16_1236_002.png

To find x, we need to know the measure of the unshaded reguon's central angle.

Then we would have two of the three angle measures of the triangle, where x is the third angle.

Find the central angle using the unshaded part's area as a portion or fraction of the circle's area.

Area of circle with radius 5 = $$25\pi$$
Area of shaded region, given: $$20\pi$$
Area of unshaded region: $$25\pi - 20\pi = 5\pi$$

Find the unshaded sector's fractional amount of the circle.

$$\frac{SectorArea}{CircleArea}=\frac{CentralAngle}{360°}$$

$$\frac{5\pi}{25\pi}=\frac{1}{5}=\frac{CentralAngle}{360°}$$

$$\frac{1}{5}$$ of 360 is 72°. The right triangle's second angle = 90°.

x = (180 - 90 - 72) = 18 degrees

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Re: The circle shown above has center O and radius of length 5. If the are [#permalink]

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16 Nov 2017, 10:51

A circle with radius 5 will have an area 25π.

Since the area of the shaded region is 20π,
the sector's area must be in a ratio of 1:4 with the shaded region.

Hence. the angle in the sector will also be $$\frac{1}{(1+4)}*360 = 72$$

We also know that the area in a triangle sums up to give 180 degree.
Therefore, x+72+90 = 180 => x = 18(Option A)
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Kudos [?]: 750 [0], given: 20

Re: The circle shown above has center O and radius of length 5. If the are   [#permalink] 16 Nov 2017, 10:51
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