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# A solid cube is placed in a cylindrical container. Which of the follow

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Intern
Joined: 29 Aug 2010
Posts: 16
A solid cube is placed in a cylindrical container. Which of the follow  [#permalink]

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29 Oct 2017, 22:47
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Difficulty:

85% (hard)

Question Stats:

55% (01:45) correct 45% (02:18) wrong based on 47 sessions

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A solid cube is placed in a cylindrical container. Which of the following percent values COULD possibly represent the ratio of the volume of the cylinder not occupied by the cube to the volume of the cylinder? (Assume the value of $$\pi$$ to be 3)

(A) 16%
(B) 25%
(C) 28%
(D) 32%
(E) 36%
Senior Manager
Joined: 05 Dec 2016
Posts: 259
Concentration: Strategy, Finance
GMAT 1: 620 Q46 V29
Re: A solid cube is placed in a cylindrical container. Which of the follow  [#permalink]

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30 Oct 2017, 00:37
Let's consider extreme situation:
Assume that cube is inscribed into the cylinder, then the ratio of its sides to radius of cylinder would be 2^1/2 to 1.
Volume of cube=(2^1/2)^3=2^3/2=2*2^1/2
Volume of cylinder=3*1^2*2^1/2=3*2^1/2
Volume of space in cylinder not occupied by cube=3*2^1/2-2*2^1/2=2^1/2
Ratio of volume not occupied by cube to volume of cylinder= 2^1/2 : 3*2^1/2=1/3

(a) 16%
Means that remaining 84% is occupied by cube, which cannot be true because maximum possible occupied space for cube would be 66,66%
(b) 25%
Means that remaining 75% is occupied by cube, which cannot be true because maximum possible occupied space for cube would be 66,66%
(c) 28%
Means that remaining 72% is occupied by cube, which cannot be true because maximum possible occupied space for cube would be 66,66%
(d) 32%
Means that remaining 68% is occupied by cube, which cannot be true because maximum possible occupied space for cube would be 66,66%
(e) 36%
Means that remaining 64% is occupied by cube, which CAN be true because maximum possible occupied space for cube would be 66,66%
Hence we can infer that cube is not inscribed in the cylinder.

Intern
Joined: 29 Aug 2010
Posts: 16
Re: A solid cube is placed in a cylindrical container. Which of the follow  [#permalink]

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30 Oct 2017, 23:25
1
1
Let us consider the situation where the largest possible cube (shaded in grey) fits perfectly in a particular cylinder as shown in the diagram below; however, the question does not state that we must fit in a largest possible cube into the cylinder.

Let the edge of the cube be $$a$$.
Since the cube has to have a perfect fit (minimum non-utilization of cylinder space),
we must have:
Height of the cylinder = edge of the cube = $$a$$

In right angled triangle CAB:
$$CB^2$$ $$=$$ $$CA^2$$+$$AB^2$$ $$=$$ $$a^2$$ + $$a^2$$ $$=$$ $$2a^2$$
$$=> CB =a\sqrt{2}$$
Thus, the diameter of the cylinder $$= a\sqrt{2}$$

=> Radius of the cylinder $$=$$$$\frac{a}{\sqrt{2}}$$

Thus, volume of the cylinder
$$=$$ $$\pi$$ X $$radius^2$$ X $$height$$
$$=$$ $$\pi$$ X $$(\frac{a}{\sqrt{2}})^2$$ X $$a$$
$$= \frac{3}{2}a^3$$

Volume of the Cube $$= a^3$$
Thus, volume of the cylinder not occupied by the cube
$$=$$ $$\frac{3}{2}$$$$a^3$$$$-$$$$a^3$$
$$= \frac{a^3}{2}$$

Thus, the required percent

$$= (\frac{a^3}{2})/(\frac{3}{2}a^3) X 100$$
$$=$$ 33.3 %

The above situation depicts the case where minimum volume of the cylinder as a percent of the
total volume of the cylinder is unutilized, i.e. not covered by the cube.
Thus, in any other scenario, the required percent value would be either greater than or equal to
33.3%

The only possible value from the answer options is 36%

The correct answer is option E.
Attachments

Cylinder.PNG [ 9.5 KiB | Viewed 582 times ]

Re: A solid cube is placed in a cylindrical container. Which of the follow &nbs [#permalink] 30 Oct 2017, 23:25
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