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10 Mar 2011, 08:50
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Be sure to select an answer first to save it in the Error Log before revealing the correct answer (OA)!
Difficulty:
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67% (00:48) correct
33%
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wrong
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Q. A rectangle with sides w and h has a circle with radius 2 cut out of it. If the missing circle is tangent to three of the rectangles edges and the ratio of w to h is 2:1, what is the area of the remaining pieces of the rectangle?
8 - 4*pi
8 - 2*pi
32 - 4*pi
24
32 - 2*pi
Think about how one would go about solving this question and how that corresponds to the way the answer choices are structured. Why does the correct answer have to be one of the "integer minus pi-term" choices? Why must the pi-term be 4*pi? What is the only logical answer?
Solution
8 - 4*pi
8 - 2*pi
32 - 4*pi
24
32 - 2*pi
Think about how one would go about solving this question and how that corresponds to the way the answer choices are structured. Why does the correct answer have to be one of the "integer minus pi-term" choices? Why must the pi-term be 4*pi? What is the only logical answer?
Solution
The question describes a rectangle with a circle cut out. So the area will be found by subtracting the area of the circle from the area of the rectangle. This is represented by the "integer minus pi-term" answer choices, since a circle with an integer radius will have an area that is a factor of pi.
Since the circle is tangent to three of the rectangle's edges it must be completely within the rectangle, which means its entire area will be subtracted from the area of the rectangle. Because it is given that the circle's radius is 2, its area is 4*pi.
Looking at the two answer choices with 4*pi terms, only one is logically valid. If 4*pi were subtracted from 8, the result would be negative, an impossibility for a geometric area on the GMAT. Thus, the only logical answer is 32 - 4*pi.
When the GMAT uses answer choices that contain multiple term expressions, often the opportunity to reverse-engineer the answers is present. This phenomena extends beyond geometry questions and occasionally can be worked the other way. Check out PS Q. 145 in the Quantitative Review, 2nd Ed. How can the expression of the area of the shaded region be used to answer the question?
Since the circle is tangent to three of the rectangle's edges it must be completely within the rectangle, which means its entire area will be subtracted from the area of the rectangle. Because it is given that the circle's radius is 2, its area is 4*pi.
Looking at the two answer choices with 4*pi terms, only one is logically valid. If 4*pi were subtracted from 8, the result would be negative, an impossibility for a geometric area on the GMAT. Thus, the only logical answer is 32 - 4*pi.
When the GMAT uses answer choices that contain multiple term expressions, often the opportunity to reverse-engineer the answers is present. This phenomena extends beyond geometry questions and occasionally can be worked the other way. Check out PS Q. 145 in the Quantitative Review, 2nd Ed. How can the expression of the area of the shaded region be used to answer the question?
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10 Mar 2011, 11:26
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I believe the three edges of the rectangle are tangents to the circle and not the vice-versa.
CF=EC=EH=AG=2
HA=EG=2*EC=4
HI=2*HA=2*4=8
Area of rectangle = 32
Area of circle = \(\pi*(2)^2=4\pi\)
Area of required portion \(= 32-4\pi\)
CF=EC=EH=AG=2
HA=EG=2*EC=4
HI=2*HA=2*4=8
Area of rectangle = 32
Area of circle = \(\pi*(2)^2=4\pi\)
Area of required portion \(= 32-4\pi\)
Attachments
circle_within_rectangle.PNG [ 10.87 KiB | Viewed 2038 times ]
10 Mar 2011, 14:11
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spacelandprep
I don't see much of a "strategy" here
You cant get away from calculating area of the circle \(4*pi\)
You need to use the fact that circle would be completely inside the rectangle and then it is not so far fetched to conclude that circle's diameter would necessarily be equal to the smaller side and hence smaller side is 4 and larger side would be 8 as ratio is given as 2:1. This will give answer as \(4*8 - 4*pi\)
So, conventional Math would give the fastest answer without any need to resort to any "strategy".
