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23 Jul 2017, 23:28
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A
Be sure to select an answer first to save it in the Error Log before revealing the correct answer (OA)!
Difficulty:
95%
(hard)
Question Stats:
34% (02:19) correct
66%
(02:08)
wrong
based on 68
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The volume of a right circular cylinder equals half the volume of a cube. Does the cylinder fit inside the cube?
(1) The ratio of height to radius of the cylinder is 2:1.
(2) Each side of the cube is less than twice the height of the cylinder.
(1) The ratio of height to radius of the cylinder is 2:1.
(2) Each side of the cube is less than twice the height of the cylinder.
24 Jul 2017, 17:52
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Bunuel
It should be A
We know
Area of Cylinder = \(πR^2H\) -- where \(R\) and \(H\) are radius and height of the Cylinder respectively.
Area of Cube = \(S^3\) -- where S is the side of the cube.
In this case, we are given
\(πR^2*H = \frac{1}{2}*S^3\)
Statement 1: Sufficient
We know \(H = 2R\) so we can replace it in the equation above.
\(πR^2*2R = \frac{1}{2}*S^3\)
\(\frac{S^3}{R^3} = 4π\)
\(\frac{S}{R} = \sqrt[3]{4π}\)
In a Square, we know that Sides and Diagonal are in the ratio \(1:\sqrt{2}\) but we know here that Diagonal \(D = 2R\)
\(\frac{S}{D} = \frac{1}{\sqrt{2}}\)
but we know \(D = 2R\)
\(\frac{S}{2R} = \frac{1}{\sqrt{2}}\)
\(\frac{S}{R} = \sqrt{2}\)
For a cube to fit inside a cylinder, its sides must be less than \(\sqrt{2}\) times the radius provided its sides do not exceed the height of the cylinder. Since we already have the ratios of radius and height, and radius and sides, we should be able to compare the two \(\frac{S}{R}\) ratios and determine whether the cube can fit inside the cylinder.
Statement 2: Insufficient
\(S < 2H\) but \(S < H\) would be more helpful information and we do not know the ratio of radius of cylinder to its height and thus have no way to find out if the cube would fit inside the cylinder.
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07 Sep 2023, 07:18
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