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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, 06:52
1
2
00:00
A
B
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D
E

Difficulty:

  25% (medium)

Question Stats:

78% (01:56) correct 22% (02:17) wrong based on 128 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) \(9\sqrt{3}\)


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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, 07:19
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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, 14:17
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, 02:33
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Re: The shaded region in the …gure above represents a circular  [#permalink]

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New post 17 Apr 2019, 06:24
You can solve this more quickly if you understand the given parameters.
We're given the ratio of the two areas and the radius of the big circle, it perfectly fits to the equation

(R/r)^2 = (S/s)
(9/r)^2 = (2/1)
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Re: The shaded region in the …gure above represents a circular   [#permalink] 17 Apr 2019, 06:24
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