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If the radius of the circle H in the figure above is 2/3 of the radius

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Bunuel
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If the radius of the circle H in the figure above is 2/3 of the radius [#permalink] New post   13 Nov 2017, 08:12
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If the radius of the circle H in the figure above is 2/3 of the radius of the circle G, and the radius of circle G is 3/4 of the radius of the circle F, then the area of the shaded region is what part of the area of circle F?

(A) 1/12
(B) 17/144
(C) 1/4
(D) 5/16
(E) 9/16

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Re: If the radius of the circle H in the figure above is 2/3 of the radius [#permalink] New post   13 Nov 2017, 08:39
Bunuel wrote:

If the radius of the circle H in the figure above is 2/3 of the radius of the circle G, and the radius of circle G is 3/4 of the radius of the circle F, then the area of the shaded region is what part of the area of circle F?

(A) 1/12
(B) 17/144
(C) 1/4
(D) 5/16
(E) 9/16

Show Spoiler
Attachment:
2017-11-13_1910.png


Let the radius of circle F be x; radius of circle G will be 3x/4; radius of circle H will be x/2
Area of shaded region is 9x^2/16 - x^2/4 = 5x^2/16 (pie)
Shaded region part is 5/16 part of area of circle F
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If the radius of the circle H in the figure above is 2/3 of the radius [#permalink] New post   13 Nov 2017, 08:43
Bunuel wrote:

If the radius of the circle H in the figure above is 2/3 of the radius of the circle G, and the radius of circle G is 3/4 of the radius of the circle F, then the area of the shaded region is what part of the area of circle F?

(A) 1/12
(B) 17/144
(C) 1/4
(D) 5/16
(E) 9/16

Show Spoiler
Attachment:
2017-11-13_1910.png


let the radius of circle F be \(12\)

so radius of circle G \(=12*\frac{3}{4}=9\) and radius of circle H \(= 9*\frac{2}{3}=6\)

Area of shaded region = Area of circle G-Area of circle H\(= π9^2-π6^2 = 45π\)

Area of circle F \(= π12^2=144π\)

Area of shaded region to area of circle F \(= \frac{45π}{144π}=\frac{5}{16}\)

Option D
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If the radius of the circle H in the figure above is 2/3 of the radius [#permalink] New post   13 Nov 2017, 10:58
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Bunuel wrote:

If the radius of the circle H in the figure above is 2/3 of the radius of the circle G, and the radius of circle G is 3/4 of the radius of the circle F, then the area of the shaded region is what part of the area of circle F?

(A) 1/12
(B) 17/144
(C) 1/4
(D) 5/16
(E) 9/16

Show Spoiler
Attachment:
2017-11-13_1910.png

Radii for H, G, and F:

\(H = \frac{2}{3}G\)
\(\frac{H}{G}=\frac{2}{3}\)
\(H = 2, G = 3\)

\(G = \frac{3}{4}F\)
\(3=\frac{3}{4}F\)
\(F = 4\)

Area of H = \(\pi*r^2= 4\pi\)
Area of G = \(9\pi\)
Area of F = \(16\pi\)

Shaded region's area = (Area of G) - (Area of H)
\((9\pi - 4\pi) = 5\pi\)

Area of the shaded region is what part of the area of circle F?
\(\frac{5\pi}{16\pi}=\frac{5}{16}\)

Answer D
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If the radius of the circle H in the figure above is 2/3 of the radius [#permalink]
13 Nov 2017, 10:58
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