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The shaded region in the figure above represents a

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The shaded region in the figure above represents a [#permalink] New post 29 Jun 2012, 14:14
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The shaded region in the figure above represents a rectangular frame with length 18 inches and width 15 inches. The frame encloses a rectangular picture that has the same area as the frame itself. If the length and width of the picture have the same ratio as the lenght and width of the frame, what is the length of the picture, in inches?

A. 9\sqrt2
B. \frac {3}{2}
C. \frac {9}{\sqrt2}
D.15 ( 1 - \frac {1}{\sqrt2}
E. \frac {9}{2}
[Reveal] Spoiler: OA

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Last edited by Bunuel on 29 Jun 2012, 16:07, edited 1 time in total.
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Re: The shaded region in the figure above represents a [#permalink] New post 29 Jun 2012, 16:06
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Stiv wrote:
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The shaded region in the figure above represents a rectangular frame with length 18 inches and width 15 inches. The frame encloses a rectangular picture that has the same area as the frame itself. If the length and width of the picture have the same ratio as the lenght and width of the frame, what is the length of the picture, in inches?

A. 9\sqrt2
B. \frac {3}{2}
C. \frac {9}{\sqrt2}
D.15 ( 1 - \frac {1}{\sqrt2}
E. \frac {9}{2}


Say the length and the width of the picture are x and y respectively. Since they have the same ratio as the lenght and width of the frame, then \frac{x}{y}=\frac{18}{15} --> y=\frac{5}{6}x.

Next, since the frame encloses a rectangular picture that has the same area as the frame itself and the whole area is 18*15, then the areas of the frame (shaded region) and the picture (inner region) are \frac{18*15}{2}=9*15 each.

The area of the picture is xy=9*15 --> x*(\frac{5}{6}x)=9*15 --> x^2=2*81 --> x=9\sqrt{2}.

Answer: A.
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Re: The shaded region in the figure above represents a [#permalink] New post 01 Jul 2012, 02:56
The total area is 15*18=270, the area of the picture is half of the whole area = 135. the ration of the width and length of the picture is the same as the frames 15/18 or 5/6. We need to find the length of the picture 5x*6x=135, 30x^2=135, x^2=135/30, x=3/sqrt2, so the length = 6*3/sqrt2=9sqrt2
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Re: The shaded region in the figure above represents a [#permalink] New post 14 Jul 2012, 01:22
Yes, I understand now. Thank you for the explanation!
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Re: The shaded region in the figure above represents a   [#permalink] 14 Jul 2012, 01:22
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