gmatter2012 wrote:The shaded region in the figure below 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*sqrt(2)
(B) 3/2
(C) 9/sqrt(2)
(D) 15(1-(1/2))
(E) 9/2
Total area = 18*15 = 270.
Since the frame and the picture have equal areas, the area of the picture = (1/2)270 = 135.
In the frame, height : width = 18:15 = 6:5.
Since the dimensions of the picture are in the same ratio, let the length of the picture = 6x and the width of the picture = 5x.
Thus:
(6x)(5x) = 135
x² = 9/2
x = 3/√2.
The length of the picture = 6x = 6(3/√2) = 18/√2 * √2/√2 = (18√2)/2 = 9√2.
The correct answer is
A.
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