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16 Nov 2022, 11:34
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B
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Difficulty:
55%
(hard)
Question Stats:
63% (03:05) correct
37%
(03:18)
wrong
based on 19
sessions
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The figure above shows a ceramic tile in the shape of an equilateral triangle with sides of length 12 inches. Parts of the tile are gray, as represented by the shaded regions, and the rest of the tile is white. The equilateral triangle in the middle of the tile has sides of length 9 inches and consists of nine congruent equilateral triangles. What fraction of the area of the tile is gray?
A. 3/10
B. 3/8
C. 2/5
D. 1/2
E. 3/5
Attachment:
Untitled.png [ 32.9 KiB | Viewed 1840 times ]
22 Nov 2022, 09:21
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Bunuel
Area of an equilateral \(\Delta \) \(= \frac{\sqrt 3}{4}*s^2\) where \(s\) is the side of the equilateral \(\Delta \).
The area of the smaller equilateral \(\Delta \) with side \(9\) inches is \(\frac{\sqrt 3}{4}*81\)
Area of each small individual equilateral \(\Delta \) within this \(9\) sided equilateral \(\Delta \) =\(\frac{\sqrt 3}{4}*\frac{81}{9} = \frac{\sqrt 3}{4}*9\)
Since there are \(6\) grey equilateral \(\Delta 's = \frac{\sqrt 3}{4}*9*6\) = area of shaded grey region.
Area of large equilateral \(\Delta \) with side \(12 =\) \(\frac{\sqrt 3}{4}*144\)
Required fraction:
\(\frac{\frac{\sqrt 3}{4}*9*6 }{\frac{\sqrt 3}{4}*144} = \frac{3}{8}\)
Ans B
Hope it's clear.
