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# In the rectangle above, the shaded region is bounded in part by 4 equa

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Math Expert
Joined: 02 Sep 2009
Posts: 52343
In the rectangle above, the shaded region is bounded in part by 4 equa  [#permalink]

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23 Nov 2017, 00:18
00:00

Difficulty:

25% (medium)

Question Stats:

100% (01:11) correct 0% (00:00) wrong based on 15 sessions

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In the rectangle above, the shaded region is bounded in part by 4 equal arcs with centers at P, Q, R and S. If PQ = 6 and QR = 10, what is the area of the shaded region?

(A) 60 – 9π
(B) 60 – 4π
(C) 32 – 9π
(D) 32 – 4π
(E) 30 – 6π

Attachment:

2017-11-23_1214.png [ 5.04 KiB | Viewed 499 times ]

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In the rectangle above, the shaded region is bounded in part by 4 equa  [#permalink]

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23 Nov 2017, 12:31
Bunuel wrote:

In the rectangle above, the shaded region is bounded in part by 4 equal arcs with centers at P, Q, R and S. If PQ = 6 and QR = 10, what is the area of the shaded region?

(A) 60 – 9π
(B) 60 – 4π
(C) 32 – 9π
(D) 32 – 4π
(E) 30 – 6π

Attachment:
2017-11-23_1214.png

The shaded region's area = (rectangle area) - (area of the four unshaded regions) inside the rectangle -- call them "sectors."

Area of rectangle = LW = (6*10) = 60

Sectors' arc lengths are equal.

Each arc's sector has a central angle of 90°, the degree measure of a rectangle's vertex. Each sector therefore = $$\frac{90°}{360°}=\frac{1}{4}$$ of a circle

Each sector's radius = $$\frac{1}{2}$$ PQ. PQ = 6.
Each sector has r = 3

Sectors have: equal arc lengths; equal central angles; and equal radii.
Hence the four sectors' areas are equal.

$$\frac{1}{4}$$ circle * 4 = 1 whole circle = total of four sector areas

Area of one circle with radius = 3: $$πr^2 = 9π$$

(Rectangle area) - (1 circle's area)

$$60 - 9π$$

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In the rectangle above, the shaded region is bounded in part by 4 equa &nbs [#permalink] 23 Nov 2017, 12:31
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