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The circumference of a circle is 10pi. Which of the followin

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The circumference of a circle is 10pi. Which of the followin  [#permalink]

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New post 19 May 2013, 00:55
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The circumference of a circle is 10pi. Which of the following is not a possible value of the area of a rectangle inscribed in it?

A. 30
B. 40
C. \(20\sqrt{2}\)
D. \(30\sqrt{2}\)
E. \(40\sqrt{2}\)

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Re: The circumference of a circle is 10pi. Which of the followin  [#permalink]

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New post 19 May 2013, 01:09
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GMATtracted wrote:
The circumference of a circle is 10pi. Which of the following is not a possible value of the area of a rectangle inscribed in it?

A. 30
B. 40
C. \(20\sqrt{2}\)
D. \(30\sqrt{2}\)
E. \(40\sqrt{2}\)


The maximum area of a rectangle inscribed in a circle is when the rectangle is a square. Now, 2pi*r = 10pi. Thus 2r = 10. Thus, as the diameter of the circle is the diagonal of the square, the side of the square = \(10/\sqrt{2}\) = \(5*\sqrt{2}\)--> Area = \(side^2\)= 50 sq. units. Thus any value from the given options more than 50 will be the answer.
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Re: The circumference of a circle is 10pi. Which of the followin  [#permalink]

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New post 29 May 2013, 07:55
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1
GMATtracted wrote:
The circumference of a circle is 10pi. Which of the following is not a possible value of the area of a rectangle inscribed in it?

A. 30
B. 40
C. \(20\sqrt{2}\)
D. \(30\sqrt{2}\)
E. \(40\sqrt{2}\)


Probably the quickest way to answer this question is to pick the biggest value from the answer. There is infinite number of rectangles that can be inscribed in a circle. For example, if a rectangle with area of 40 fits in the circle, rectangle with area of 30 will fit as well. If answer D fits, A, B, and C all fit. In that case, if there is only one answer, like in GMAT, that will be E.
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Re: The circumference of a circle is 10pi. Which of the followin  [#permalink]

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New post 30 May 2013, 16:54
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Step 1: the maximum side of the rectangle must be less than 2*radius (diameter). If the radius is 5 (as perimeter=2*pi*r=10*pi), the maximum side of the inscribed rectangle must be less than 10.

Step 2: let's try to find the maximum area of an inscribed rectangle: thinking a little bit, it can be found that the maximum area will be when we pick an inscribed square (see the red square in the drawing). If each black arrow (radius) measure \(5\), the sides of this red square have to be \(\sqrt{2}*5\). Why? Because \(45\) - \(45\) - \(90\) triangles have sides \(X\) - \(X\) - \(\sqrt{2}*X\).

Step 3: therefore, the maximum area is \((\sqrt{2}*5)^2=25*2=50\)

SOLUTION: \(40*\sqrt{2}=40*1.41...=56.5...\) the only possible solution is E
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Re: The circumference of a circle is 10pi. Which of the followin  [#permalink]

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New post 28 Feb 2017, 07:16
Posting official solution of this problem.
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Re: The circumference of a circle is 10pi. Which of the followin  [#permalink]

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New post 07 Apr 2018, 13:59
The important piece of this question is to know that every square is a special kind of rectangle, but not every rectangle is a square. The question would have been easier if it mentioned "quadrilateral" instead of rectangle.
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Re: The circumference of a circle is 10pi. Which of the followin &nbs [#permalink] 07 Apr 2018, 13:59
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