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Re: The shaded an ar in the figure over represents a rectangular frame<#permalink>18 Sep 2017, 13:26
The shaded region in the figure over represents a rectangular frame with length 18 inches and width 15 inches. The structure encloses a rectangular snapshot that has the same area together the structure itself. If the length and width the the picture have the same proportion as the lenght and also width that the frame, what is the length of the picture, in inches?A. (9sqrt2)B. (frac 32)C. (frac 9sqrt2)D. (15 ( 1 - frac 1sqrt2)E. (frac 92)
Say the length and also the width of the photo are (x) and (y) respectively. Since they have actually the same ratio as the lenght and width of the frame, climate (fracxy=frac1815) --> (y=frac56x).Next, since the frame encloses a rectangular picture that has actually the same area as the frame itself and also the whole area is (18*15), climate the locations of the structure (shaded region) and also the photo (inner region) are (frac18*152=9*15) each.The area that the snapshot is (xy=9*15) --> (x*(frac56x)=9*15) --> (x^2=2*81) --> (x=9sqrt2).Answer: A.

Re: The shaded an ar in the figure above represents a rectangle-shaped frame<#permalink>18 Sep 2017, 20:32
The shaded region in the figure over represents a rectangular structure with size 18 inches and also width 15 inches. The structure encloses a rectangular snapshot that has actually the exact same area as the framework itself. If the length and also width that the photo have the same ratio as the lenght and width the the frame, what is the length of the picture, in inches?A. (9sqrt2)B. (frac 32)C. (frac 9sqrt2)D. (15 ( 1 - frac 1sqrt2)E. (frac 92)
Say the length and the broad of the photo are (x) and also (y) respectively. Since they have actually the same proportion as the lenght and width the the frame, then(fracxy=frac1815) --> (y=frac56x).Next, because the structure encloses a rectangular snapshot that has the same area as the structure itself and also the entirety area is (18*15), climate the areas of the structure (shaded region) and also the snapshot (inner region) are (frac18*152=9*15) each.The area that the picture is (xy=9*15) --> (x*(frac56x)=9*15) --> (x^2=2*81) --> (x=9sqrt2).Answer: A.
In (xy=9*15), we substitute y in terms of x, which we found over (check the emphasize part) to acquire (x*(frac56x)=9*15). This allows us to obtain an equation with just one change x, and also solve it._________________
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Re: The shaded region in the figure over represents a rectangular frame<#permalink>20 Sep 2018, 14:11
The shaded region in the figure above represents a rectangular frame with size 18 inches and width 15 inches. The structure encloses a rectangular photo that has the same area together the framework itself. If the length and width that the picture have the same ratio as the lenght and width the the frame, what is the size of the picture, in inches?A. (9sqrt2)B. (frac 32)C. (frac 9sqrt2)D. (15 ( 1 - frac 1sqrt2))E. (frac 92)
From the concern stem ("...the length and width the the picture have the same proportion as the length and width the the frame") us know: The frame+picture ("big" rectangle) and also the snapshot ("small" rectangle) are two comparable rectangles. (*)(*) From over we have proportionality on the corresponding sides. The necessary added condition - equality in the matching internal angle - is guaranteed: they room all 90 degrees!Again from the question stem we recognize what the examiner defines as "length" and "width" (by the dimensions connected to these words), so the our emphasis is: (? = x,,,,left< extinches ight>,,,,,,left( extSee,, extfigure,, extbelow ight)) from "The frame encloses a rectangular snapshot that has the same area together the structure itself." we know that the "big" (rectangle) has actually TWICE the area of the "small" (rectangle). To protect against using the second dimension that the picture, together it was done in vault (correct) solutions, let´s remember an essential geometric property: In any two similar polygons, the ratio of their locations is same to the square the the ratio of similarity that the polygons! Therefore: (2 = fracS_, extbigS_, extsmall = left( frac18x ight)^2,,,,mathop Rightarrow limits^x,, > ,,0 ,,,,,sqrt 2 = frac18x,,,,,,, Rightarrow ,,,,,xsqrt 2 = 18)(xsqrt 2 = 18,,,,,, Rightarrow ,,,,,xsqrt 2 cdot sqrt 2 = 18sqrt 2 ,,,,,, Rightarrow ,,,,,? = x = 9sqrt 2)This solution complies with the notations and also rationale taught in the GMATH method. Regards, Fabio.
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