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13 Dec 2017, 23:33
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D
Be sure to select an answer first to save it in the Error Log before revealing the correct answer (OA)!
Difficulty:
55%
(hard)
Question Stats:
68% (02:51) correct
32%
(02:15)
wrong
based on 96
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In the figure, AB is parallel to CD and \(AB=CD=2 \sqrt{2}\). If the distance between AB and CD is \(2\sqrt{2}\), what is the area of the shaded region?
(A) \(2+\pi\)
(B) \(2+2\pi\)
(C) \(4+\pi\)
(D) \(4+2\pi\)
(E) \(4+4\pi\)
Attachment:
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14 Dec 2017, 15:25
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Bunuel
Attachment:
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I solved the first time in a more traditional way. My answer had subtraction.
I realized that the only way to get a plus sign was to include subtracting another subtraction.
So: Area of shaded = (area of circle) - (area of two unshaded regions)
Find the circle's area
Connect A to C, and B to D.
That is a square with four equal sides whose length = \(2\sqrt{2}\)
Given: AB = CD = \(2\sqrt{2}\)
Given: distance between AB and CD [which are parallel] is \(2\sqrt{2}\), so side AC = BD = \(2\sqrt{2}\)
The diagonal of the square:
\(s\sqrt{2}=(2\sqrt{2}*\sqrt{2})=4\)
Diagonal = diameter of circle, \(d\)
\(4 = d = 2r\)
\(r = 2\)
Area of circle: \(\pi r^2 = 4\pi\)
Find the area of the UNshaded regions
If the shaded area were just the square:
Each unshaded region would have an area calculated by (circle area - square area)/4
Area of the drawn square ABCD = \(s^2 = (2\sqrt{2})^2 = 8\)
\(\frac{4\pi - 8}{4} = (\pi - 2)\) = area of each unshaded region
OR
The square's diagonals are perpendicular bisectors, such that
Sectors' central angles are 90
Each sector's total area \(= \pi\)
-- \(\frac{90}{360} = \frac{1}{4} =\frac{?}{4\pi} =\)
\(\pi =\) sector area
The area of the triangle in each sector (where sides = radii = 2) is \(\frac{2 * 2}{2} = 2\)
So an UNshaded region in a sector would have area calculated by (Sector area - triangle area), namely, \((\pi - 2)\)
Find the shaded area
There are two sectors that contain unshaded regions. The area of each UNshaded region \(= (\pi - 2)\).
There are two sector areas, fully shaded, whose shaded area is simply \(\frac{1}{4}\) of the circle's area: \(\pi\).
Ignore the two fully shaded sectors (because if you do not, as far as I can tell, your answers will look nothing like the choices).
To find the shaded area, subtract the area of the unshaded regions from the whole circle's area.
In other words, use (circle area) - (area of unshaded region in 2 of the sectors)
(Total circle area) - (area of TWO unshaded regions) = shaded area
\(4\pi - (\pi - 2) - (\pi - 2) =\)
\(4\pi - \pi + 2 -\pi + 2 =\)
\(2\pi + 4\)
Answer
D
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General Discussion
05 Jun 2020, 06:34
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Area of the square is 8. So, the area of the shaded region would be a little more than 8. Choice D fits my guess. So, it is D.
