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A cylinder is placed inside a cube so that it stands upright when the

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A cylinder is placed inside a cube so that it stands upright when the  [#permalink]

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New post 28 Sep 2016, 01:45
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A cylinder is placed inside a cube so that it stands upright when the cube rests on one of its faces. If the volume of the cube is 16, what is the maximum possible volume of the cylinder that fits inside the cube as described?

A. 16/π
B. 2π
C. 8
D. 4π
E. 8π

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Re: A cylinder is placed inside a cube so that it stands upright when the  [#permalink]

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New post 28 Sep 2016, 17:46
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Volume of cube = \(s^{3}\) = 16
Volume of cylinder = \(π r^{2} h\)
To get maximum volume, we need to maximize \(r^{2}\)
Let the cylinder rest on one of the sides and the base fits completely on the surface.

We have Diameter of cylinder = side of cube
2r = s
Let the cylinder fit in he cube vertically, maximizing the height.
h = s

Volume = \(π \frac{(s}{2)}^{2} s\)
= \(\frac{π}{4} s^{3}\)
=\(\frac{π}{4} * 16\)
= \(4π\)
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Re: A cylinder is placed inside a cube so that it stands upright when the  [#permalink]

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New post 26 Apr 2017, 14:56
volume of cube is 16.
that means that a side of a cube is 3rd radical of 16, or 16 ^ (1/3)
volume of a cylinder is pi * r^2 * h
h is side of the cube
r is half of the side - [16 ^ (1/3)]/2

r^2 = 16^ (2/3)/4
we now have everything we need:
r^2 * h = [16^(2/3) * 16^(1/3)]/4 -> exponents are added and we get 16^1, which is 16.
16/4 = 4.
4*pi = 4pi.
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Re: A cylinder is placed inside a cube so that it stands upright when the  [#permalink]

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New post 26 Apr 2017, 19:26
Great Question.
Here is what i did =>
Let the side of cube =x

x^3=16

Radius of cylinder => x/2
Height => x

Hence Volume => π*r^2*h=> π*(x^2/4)*x => π*x^3/4
Putting x^3=16 we get => π*16/4 => 4π


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Re: A cylinder is placed inside a cube so that it stands upright when the  [#permalink]

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New post 30 Apr 2017, 13:24
I solved this in a pretty janky way.

I took the cubed root of 16 to be 5/2 = side of the cube and diameter of the cylinder.
Volume of a cylinder = pi (r^2) h
V = pi (5/2 * 1/2)^2 * (5/2) ... Note: the height of the cylinder is equal to the side of the cube, so again = 5/2
V = 125/32 (pi) = approximately 4 pi.
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Re: A cylinder is placed inside a cube so that it stands upright when the  [#permalink]

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New post 13 Jun 2017, 18:12
Well, i just looked at this question literally,
Considering that maximum volume will be achieved if the measures of the cylinder (the area of circular base and the height) need to be as close to that of the cube, in such a way that i could touch upper and lover surface and its sides heights are touch the surface the cube, then the cylinder can not have more than 16 unknown unit. Therefore, out of the options provided only option D, 4 pi was close enough to 16.
Had it been option C, then there would be enough space of two cylinder.
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Re: A cylinder is placed inside a cube so that it stands upright when the  [#permalink]

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New post 20 Jun 2017, 06:37
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Bunuel wrote:
A cylinder is placed inside a cube so that it stands upright when the cube rests on one of its faces. If the volume of the cube is 16, what is the maximum possible volume of the cylinder that fits inside the cube as described?

A. 16/π
B. 2π
C. 8
D. 4π
E. 8π


The cylinder of the maximum volume that can be inscribed in a cube is one with the diameter of its base being the side length of the cube and the height also being the side length of the cube. Recall that the volume of a cylinder is V = πr^2h, where r is the radius and h is the height. If s = side length of the cube, we have r = s/2 (since s is also the length of the diameter) and h = s. Thus, the maximum volume of the cylinder is:

V = π*(s/2)^2*(s)

V = π(s^3)/4

Notice that s^3 is the volume of the cube and it’s given to be 16; thus, the maximum volume of the cylinder is:

V = π(16)/4

V = 4π

Answer: D
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Re: A cylinder is placed inside a cube so that it stands upright when the  [#permalink]

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New post 13 Dec 2017, 07:00
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Bunuel wrote:
A cylinder is placed inside a cube so that it stands upright when the cube rests on one of its faces. If the volume of the cube is 16, what is the maximum possible volume of the cylinder that fits inside the cube as described?

A. 16/π
B. 2π
C. 8
D. 4π
E. 8π


The volume of the cube is 16
Volume of cube = (side length)³
So: 16 = (side length)³
So, side length = ∛16

So, the BASE of the cube is a SQUARE with dimension ∛16 by ∛16
So, the largest cylinder to fit inside the cube must have a diameter of ∛16
This means the RADIUS of the cylinder = ∛16/2

Also, since the cube has HEIGHT ∛16, the largest cylinder to fit inside the cube must have a HEIGHT of ∛16

What is the maximum possible volume of the cylinder that fits inside the cube as described?
Volume = π(radius)²(height)
= π(∛16/2)²(∛16)
= π(16/4)
= 4π
= D

Cheers,
Brent
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