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You are watching: What is the shaded area in the figure represent?   Re: The shaded region in the figure above represents a rectangle-shaped frame<#permalink>18 Sep 2017, 13:26
The shaded an ar in the figure above represents a rectangular structure with length 18 inches and also width 15 inches. The frame encloses a rectangular picture that has actually the exact same area together the structure itself. If the length and width of 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)
Say the length and the broad of the picture are (x) and (y) respectively. Since they have actually the same proportion as the lenght and also width the the frame, climate (fracxy=frac1815) --> (y=frac56x).Next, due to the fact that the structure encloses a rectangular picture that has the exact same area as the frame itself and the totality area is (18*15), climate the areas of the frame (shaded region) and the picture (inner region) room (frac18*152=9*15) each.The area that the photo is (xy=9*15) --> (x*(frac56x)=9*15) --> (x^2=2*81) --> (x=9sqrt2).Answer: A.     Re: The shaded an ar in the figure over represents a rectangular frame<#permalink>18 Sep 2017, 20:32
The shaded region in the figure above represents a rectangular frame with length 18 inches and width 15 inches. The structure encloses a rectangular picture that has the same area as the frame itself. If the length and also width of the picture have the same proportion 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)
Say the length and also the broad of the snapshot are (x) and (y) respectively. Because they have the same proportion as the lenght and width the the frame, then(fracxy=frac1815) --> (y=frac56x).Next, since the structure encloses a rectangular snapshot that has the exact same area together the frame itself and the entirety area is (18*15), then the areas of the framework (shaded region) and the photo (inner region) space (frac18*152=9*15) each.The area the the snapshot 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 us found over (check the emphasize part) to gain (x*(frac56x)=9*15). This enables us to acquire an equation with just one variable x, and solve it._________________
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Re: The shaded region in the figure above represents a rectangle-shaped frame<#permalink>20 Sep 2018, 14:11
The shaded region in the figure over represents a rectangular framework with size 18 inches and also width 15 inches. The framework encloses a rectangular photo that has the same area as the structure itself. If the length and width of the snapshot have the same proportion as the lenght and width of 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 of the snapshot have the same ratio as the length and also width of the frame") us know: The frame+picture ("big" rectangle) and the picture ("small" rectangle) space two similar rectangles. (*)(*) From above we have proportionality top top the corresponding sides. The necessary additional condition - equality in the equivalent internal angle - is guaranteed: they are all 90 degrees!Again indigenous the question stem we understand what the examiner specifies as "length" and "width" (by the dimensions associated to this words), so that our emphasis is: (? = x,,,,left< extinches ight>,,,,,,left( extSee,, extfigure,, extbelow ight)) from "The framework encloses a rectangular photo that has actually the very same area as the framework itself." we know that the "big" (rectangle) has actually TWICE the area of the "small" (rectangle). To avoid using the second dimension that the picture, as it was done in vault (correct) solutions, let´s remember an important geometric property: In any two comparable polygons, the ratio of their areas is same to the square that the proportion of similarity the 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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