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Math Expert V
Joined: 02 Sep 2009
Posts: 55277
The volume of a right circular cylinder equals half the volume of a  [#permalink]

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Difficulty:   75% (hard)

Question Stats: 38% (02:51) correct 62% (01:59) wrong based on 30 sessions

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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.

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Joined: 14 Oct 2015
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The volume of a right circular cylinder equals half the volume of a  [#permalink]

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Bunuel wrote:
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.

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.

$$πR^2*2R = \frac{1}{2}*S^3$$

$$\frac{S^3}{R^3} = 4π$$

$$\frac{S}{R} = \sqrt{4π}$$

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.
Attachments cylindersquare.jpg [ 28.41 KiB | Viewed 840 times ]

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There is one fix to every problem, working harder! The volume of a right circular cylinder equals half the volume of a   [#permalink] 24 Jul 2017, 17:52
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# The volume of a right circular cylinder equals half the volume of a

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