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# In the figure above, the circles touch each other and touch the sides

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Math Expert
Joined: 02 Sep 2009
Posts: 42652

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In the figure above, the circles touch each other and touch the sides [#permalink]

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16 Nov 2017, 21:50
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Question Stats:

82% (00:46) correct 18% (01:03) wrong based on 32 sessions

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In the figure above, the circles touch each other and touch the sides of the rectangle at the lettered points shown. The radius of each circle is 1. Of the following, which is the best approximation to the area of the shaded region?

(A) 6
(B) 4
(C) 3
(D) 2
(E) 1

[Reveal] Spoiler:
Attachment:

2017-11-17_0946.png [ 7.88 KiB | Viewed 651 times ]
[Reveal] Spoiler: OA

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Kudos [?]: 135981 [0], given: 12719

Director
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GMAT 1: 630 Q47 V29
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Re: In the figure above, the circles touch each other and touch the sides [#permalink]

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16 Nov 2017, 23:00
Bunuel wrote:

In the figure above, the circles touch each other and touch the sides of the rectangle at the lettered points shown. The radius of each circle is 1. Of the following, which is the best approximation to the area of the shaded region?

(A) 6
(B) 4
(C) 3
(D) 2
(E) 1

[Reveal] Spoiler:
Attachment:
2017-11-17_0946.png

Length of BC = BF = 2

Area of square BCFE = 4

Area of Circle (unshaded part which overlaps Square) = 2 * 1/2 * pie * r^2 = pie

Shaded region area = 4- pie = ~1

E
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Kudos [?]: 181 [0], given: 139

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Joined: 22 May 2016
Posts: 1140

Kudos [?]: 408 [0], given: 648

Re: In the figure above, the circles touch each other and touch the sides [#permalink]

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17 Nov 2017, 13:39
Bunuel wrote:

In the figure above, the circles touch each other and touch the sides of the rectangle at the lettered points shown. The radius of each circle is 1. Of the following, which is the best approximation to the area of the shaded region?

(A) 6
(B) 4
(C) 3
(D) 2
(E) 1

Attachment:

mmmm.png [ 17.04 KiB | Viewed 305 times ]

The area of the shaded region equals
(Rectangle area) - (circles' area)/2

Rectangle area: L * W
The circles' radii completely span the length and width of the rectangle
Radius of one circle = 1

LENGTH = 4 radii = 4
WIDTH = 2 radii = 2
Rectangle's area = 4 * 2 = 8

Area of both circles, where r = 1: $$2(πr^2) = 2π$$

The "extra" area between the rectangle's edges and the circles' edges consists of 8 equal regions. See diagram.

To get total extra area, subtract circles' area from rectangle's area.
To get half the total extra area (shaded region), subtract circles' area from rectangle's area and divide by 2.

Half of extra area = area of shaded region:

(rectangle area, R) - (circles' area, C) divided by 2:

$$\frac{(R - C)}{2}$$

$$\frac{(8 - 2π)}{2}$$

$$π\approx{3.14}$$

$$\frac{(8 - 6.28)}{2}= \frac{1.72}{2} = .86$$

$$0.86\approx{1}$$

Kudos [?]: 408 [0], given: 648

Manager
Joined: 14 Sep 2016
Posts: 57

Kudos [?]: 14 [0], given: 119

Concentration: Finance, Economics
Re: In the figure above, the circles touch each other and touch the sides [#permalink]

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17 Nov 2017, 15:09
Bunuel wrote:

In the figure above, the circles touch each other and touch the sides of the rectangle at the lettered points shown. The radius of each circle is 1. Of the following, which is the best approximation to the area of the shaded region?

(A) 6
(B) 4
(C) 3
(D) 2
(E) 1

[Reveal] Spoiler:
Attachment:
The attachment 2017-11-17_0946.png is no longer available

Inside square has a an area of 4.
Subtract 2 half pokeballs from 4, so subtract one full pokeball.
Square Area 4 - Pokeball area of 3.1 = .9 shaded or E.
Attachments

poke.png [ 17.4 KiB | Viewed 274 times ]

Kudos [?]: 14 [0], given: 119

Target Test Prep Representative
Status: Founder & CEO
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Re: In the figure above, the circles touch each other and touch the sides [#permalink]

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20 Nov 2017, 11:30
Bunuel wrote:

In the figure above, the circles touch each other and touch the sides of the rectangle at the lettered points shown. The radius of each circle is 1. Of the following, which is the best approximation to the area of the shaded region?

(A) 6
(B) 4
(C) 3
(D) 2
(E) 1

[Reveal] Spoiler:
Attachment:
2017-11-17_0946.png

We can see that the area of the shaded region is half the area of the region inside of the rectangle but outside of the two circles. Thus, the area of the shaded region is half the difference between the area of the rectangle and the total area of the two circles.

Area of the rectangle = 2 x 4 = 8

Area of a circle = π x 1^2 = π

Area of shaded region = ½ x (8 - 2π) = 4 - π

Since π is approximately 3.14, the area of the shaded region ≈ 4 - 3.14 = 0.86 ≈ 1.

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Kudos [?]: 1024 [0], given: 3

Re: In the figure above, the circles touch each other and touch the sides   [#permalink] 20 Nov 2017, 11:30
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