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

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

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New post 16 Nov 2017, 01:46
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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

Attachment:
2017-11-16_1236_002.png
2017-11-16_1236_002.png [ 5.5 KiB | Viewed 2231 times ]

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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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New post 16 Nov 2017, 04:25
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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The circle shown above has center O and radius of length 5. If the are [#permalink]

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New post 16 Nov 2017, 11:17
Bunuel wrote:
Image
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

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

Answer A
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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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New post 16 Nov 2017, 11:51
Image

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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Re: The circle shown above has center O and radius of length 5. If the are   [#permalink] 16 Nov 2017, 11:51
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