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Math Expert V
Joined: 02 Sep 2009
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If the largest possible cube with volume x is enclosed in a cylinder,  [#permalink]

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Difficulty:   75% (hard)

Question Stats: 37% (01:44) correct 63% (01:46) wrong based on 56 sessions

### HideShow timer Statistics If the largest possible cube with volume x is enclosed in a cylinder, what is the volume of the cylinder?

(1) The area of the base of the cylinder is 8π.
(2) x is 64

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Re: If the largest possible cube with volume x is enclosed in a cylinder,  [#permalink]

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for the cube to be the largest possible, the diameter of the base has to be equal to the diagonal of one of the sides of the cylinder.
this makes it so that looked at from the top you would see a circumscribed circle around a square.

in order to know the volume of the cylinder you need to know the height of the cylinder and the radius or diameter of the base of the cylinder (or be able to determine them)

(1) The area of the base of the cylinder is 8π.
we can calculate the radius, but we know nothing about the height
Not suff
(2) x is 64
if the total volum of the square is 64 that makes the sides 4 and that would allow us to calculate the diameter or radius from there but again nothing is mentioned about the height

both together, we still don't know anything about the height:
E
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If the largest possible cube with volume x is enclosed in a cylinder,  [#permalink]

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aeon86 wrote:
for the cube to be the largest possible, the diameter of the base has to be equal to the diagonal of one of the sides of the cylinder.
this makes it so that looked at from the top you would see a circumscribed circle around a square.

in order to know the volume of the cylinder you need to know the height of the cylinder and the radius or diameter of the base of the cylinder (or be able to determine them)

(1) The area of the base of the cylinder is 8π.
we can calculate the radius, but we know nothing about the height
Not suff
(2) x is 64
if the total volum of the square is 64 that makes the sides 4 and that would allow us to calculate the diameter or radius from there but again nothing is mentioned about the height

both together, we still don't know anything about the height:
E

A cube has all sides equal.
therefore, if volume is x, then base=length=height=x^(1/3)
and height of cylinder = height of cube = x^(1/3)

Now diameter of cylinder = diagonal of cube = sqrt[ L*L + B*B ] = sqrt[ x^(2/3) + x^(2/3) ]
Thus, Radius = sqrt[ x^(2/3) + x^(2/3) ] / 2

Area of base of cylinder = pi* r*r = [ pi *x^(2/3)*2 ] / 4 ------- EQ. A
volume of cylinder = area of base * height of cylinder = [ pi* x^(2/3)*x^(1/3) ] / 2 = [ pi*x ]/2

So we just need to find the value of x =?

Stmt 1 - we can calculate x from EQ A. we get x=64.
Stmt 2 - we are given x = 64

So volume = pi *(64/2) = 32*pi.
Hence, both the statements are sufficient individually. Correct answer is D.
IESE School Moderator S
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Re: If the largest possible cube with volume x is enclosed in a cylinder,  [#permalink]

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You’re assuming the enclosed cylinder height exactly matches the cube height. A cube can be enclosed in a cylinder that is 3x or 100x as high as the cube.

Enclosed doesn’t have the same connotation as inscribed .

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Re: If the largest possible cube with volume x is enclosed in a cylinder,  [#permalink]

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aeon86 wrote:
You’re assuming the enclosed cylinder height exactly matches the cube height. A cube can be enclosed in a cylinder that is 3x or 100x as high as the cube.

Enclosed doesn’t have the same connotation as inscribed .

Posted from my mobile device

Yes. I assume that the question says LARGEST POSSIBLE cube which means max percentage volume of the cylinder should be covered by cube . This is only possible when height is same as that of cube and diagonal = diameter of the cylinder.
IESE School Moderator S
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Re: If the largest possible cube with volume x is enclosed in a cylinder,  [#permalink]

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I do believe you’re interpreting the question wrong then.
Keep I. Mind this is supposed to be solved in under 2 minutes, and even more it’s a 600-700 question.

If you take any cylinder the largest cube that fits will be the one where the diagonal of a side equals the diameter of the cylinder.

Also : OA is E

I do agree that if your interpretation was correct, it would be D
(And it would be much easier to calculate using 1 instead of x in that case)

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Re: If the largest possible cube with volume x is enclosed in a cylinder,  [#permalink]

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In that case we are not even given the radius of the cylinder.
A cylinder of height =1cm
Radius= 2cm, can also fit a cube(largest possible) of side 1cm.

You are pretending it wrongly because neither we are given height nor radius and therefore number of possibilities are there to fit a cube in a cylinder.

In order to fit the largest possible cube, it should cover max volume of the cylinder therefore height will be equal to side of the cube.

Also not always OA are correct.
Moreover, look at the question pattern, gmat always give questions in DS from which you can derive same values of any variable.
Stmt 1(on solving) and 2 both say x=64.
Also solving this question took me 1 min 10 sec, by my way.

It would be better if someone can share the source of the question.

Posted from my mobile device Re: If the largest possible cube with volume x is enclosed in a cylinder,   [#permalink] 19 Sep 2018, 14:35
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