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# If the area of the shaded region in the figure above is 24π, what is

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Math Expert
Joined: 02 Sep 2009
Posts: 50077
If the area of the shaded region in the figure above is 24π, what is  [#permalink]

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28 Nov 2017, 21:10
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Difficulty:

35% (medium)

Question Stats:

79% (01:36) correct 21% (01:17) wrong based on 36 sessions

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If the area of the shaded region in the figure above is 24π, what is the radius r of the smaller circle?

(A) 2
(B) 4
(C) 5
(D) 6
(E) 10

Attachment:

2017-11-29_0805_001.png [ 5.75 KiB | Viewed 705 times ]

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If the area of the shaded region in the figure above is 24π, what is  [#permalink]

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29 Nov 2017, 11:53
Bunuel wrote:

If the area of the shaded region in the figure above is 24π, what is the radius r of the smaller circle?

(A) 2
(B) 4
(C) 5
(D) 6
(E) 10

Attachment:
2017-11-29_0805_001.png

Shaded region's area = $$24\pi$$
Small circle's area = $$\pi r^2$$
Large circle's area = $$\pi (r + 2)^2$$

Area of the shaded region =
(Large circle's area) - (Small circle's area)

$$\pi (r + 2)^2 - (\pi r^2) = 24\pi$$

$$\pi (r^2 + 4r + 4) - (\pi r^2) = 24\pi$$

$$\pi r^2 + 4\pi r + 4\pi - \pi r^2 = 24\pi$$

$$4\pi r + 4\pi = 24\pi$$

$$4\pi r = 20\pi$$

$$r = 5$$

Check:
Small circle's area, with r = 5, = $$25 \pi$$
Large circle's area, with r = 7, = $$49\pi$$
Large - Small = shaded region
$$49\pi - 25\pi = 24\pi$$. That works.

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Re: If the area of the shaded region in the figure above is 24π, what is  [#permalink]

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04 Dec 2017, 11:54
1
Bunuel wrote:

If the area of the shaded region in the figure above is 24π, what is the radius r of the smaller circle?

(A) 2
(B) 4
(C) 5
(D) 6
(E) 10

Attachment:
2017-11-29_0805_001.png

We can use the formula:

((2+r)^2 - r^2)π = 24π

4 + 4r + r^2 - r^2 = 24

4 + 4r = 24

4r = 20

r = 5

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Re: If the area of the shaded region in the figure above is 24π, what is &nbs [#permalink] 04 Dec 2017, 11:54
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