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The shaded region in the …gure above represents a circular

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The shaded region in the …gure above represents a circular  [#permalink]

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New post 31 Mar 2012, 05:52
1
1
00:00
A
B
C
D
E

Difficulty:

  25% (medium)

Question Stats:

80% (01:53) correct 20% (02:19) wrong based on 149 sessions

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The shaded region in the …gure above represents a circular tire. If the distance from point O to point T is 18 inches and the area of the shaded region is equal to the area of the unshaded region, what is the radius of the unshaded region, in inches?

(A) \(\frac{9}{2}\sqrt{2}\)

(B) \(\frac{9}{2}\sqrt{2}\)

(C) 9

(D) \(9\sqrt{2}\)

(E) \(9\sqrt{3}\)


Attachment:
Circle.GIF
Circle.GIF [ 5.99 KiB | Viewed 3959 times ]

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Re: The shaded region in the …gure above represents a circular  [#permalink]

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New post 31 Mar 2012, 06:19
1
enigma123 wrote:
The shaded region in the …gure above represents a circular tire. If the distance from point O to point T is 18 inches and the area of the shaded region is equal to the area of the unshaded region, what is the radius of the unshaded region, in inches?

(A) \(\frac{9}{2}\sqrt{2}\)
(B) \(\frac{9}{2}\sqrt{2}\)
(C) 9
(D) 9\(\sqrt{2}\)
(E) 9\(\sqrt{3}\)

How to solve?


The area of the big circle is \(\pi{R^2}=324\pi\);
The area of the small circle (the unshaded region) is \(\pi{r^2}\);
The area of the shaded region is \(324\pi-\pi{r^2}\);

We are told that the area of the shaded region is equal to the area of the unshaded region, so \(\pi{r^2}=324\pi-\pi{r^2}\) --> \(2\pi{r^2}=324\pi\) --> \(r^2=162\) --> \(r=9\sqrt{2}\).

Answer: D.
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Re: The shaded region in the …gure above represents a circular  [#permalink]

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New post 31 Mar 2012, 13:17
Outer ring's area = Bigger circle area - inner circle area = pi*R_out ^2 - pi*R_in ^2
it also equals to the inner circle area, by the question stem: pi*R_out ^2 - pi*R_in ^2 = pi*R_in ^2

So:
2*pi*R_in^2 = pi R_out^2
R_in^2 = (R_out^2)/2 => (18^2)/2
R_in = sqrt[ (18^2)/2 ] = 18/sqrt(2) = 9*sqrt(2) ===> answer D.
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Re: The shaded region in the …gure above represents a circular  [#permalink]

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New post 06 Mar 2014, 01:33
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Re: The shaded region in the …gure above represents a circular  [#permalink]

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New post 28 Oct 2017, 19:34
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Re: The shaded region in the …gure above represents a circular &nbs [#permalink] 28 Oct 2017, 19:34
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