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09 Apr 2021, 12:00
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B
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Difficulty:
65%
(hard)
Question Stats:
63% (02:56) correct
37%
(03:14)
wrong
based on 46
sessions
History
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Not Attempted Yet
A circular region centered at C is inscribed in equilateral triangle ABC above. If the area of ΔABC is 12, then what is the area of the shaded region?
A. \(6-3π\)
B. \(6-π√3\)
C. \(12-π√3\)
D. \(12-3π\)
E. \(3+π√3\)
Attachment:
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16 Apr 2021, 12:15
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Bunuel
One could also use following approach and solve this question in 30 seconds:
The area of the shaded region = 1/2*(the area of the triangle - the area of the sector) = 1/2*(12 - something with π) = 6 - something with π.
Only A and B fits this format, but A is negative, so we can discard it.
Also:
C: 12 - π√3 = ~7 too big for the area of the tiny shaded region (more than half of the area of the triangle).
D: 12 - 3π = ~3 too big for the area of the tiny shaded region (~1/4 of the area of the triangle).
E: 3 + π√3 = ~8 too big for the area of the tiny shaded region (more than half of the area of the triangle).
Answer: B.
General Discussion
09 Apr 2021, 21:56
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A circular region centered at C is inscribed in equilateral triangle ABC above. If the area of ΔABC is 12, then what is the area of the shaded region?
A couple of options can be eliminated based on the visual inspection. As the shaded region is too small compared to the area of the triangle, the relatively bigger(in comparison to area of triangle) values would get sticked off.
A. \(6-3π\)
This is a negative value. Eliminate.
B. \(6-π√3\)
~0.5. A probable candidate
C. \(12-π√3\)
More than half of the area of triangle. Eliminate
D. \(12-3π\)
~3.0. Shaded region doesn't look like about \(\frac{1}{4}th\) of triangle's area. Eliminate.
E. \(3+π√3\)
~8.0. Eliminate.
Solving for the sake of doing it.
Area of equilateral triangle = \(\frac{√3}{4}x^2 = 12\) where x is the length of sides of that triangle.
\(\implies x^2 = \frac{48}{√3}\)
Perpendicular from C to AB divides AB in half at point D. Applying pythagorean theorem, we have CD = \(\frac{√3}{2}x\).
Area of circular region = \((\frac{√3}{2})^2*x^2*π*\frac{60}{360}\)
= \(π2√3\)
Area of shaded region = \(\frac{12 - 2√3π}{2} = 6 - π√3\)
Answer B.
