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555-605 (Medium)|   Geometry|      
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Bunuel
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The largest possible cube is enclosed in a cylinder and has volume x. 29 Aug 2018, 01:36
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35% (medium)

Question Stats:

64% (01:07) correct 36% (01:11) wrong based on 47 sessions

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The largest possible cube is enclosed in a cylinder and has volume x. What is the volume of the cylinder?

(1) x is 27.
(2) The height of the cylinder is 5 units.
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C
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The largest possible cube is enclosed in a cylinder and has volume x. 29 Aug 2018, 10:16
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Bunuel
The largest possible cube is enclosed in a cylinder and has volume x. What is the volume of the cylinder?

(1) x is 27.
(2) The height of the cylinder is 5 units.
Show more

Refer the top view in the enclosed diagram(When a cube is inserted in a cylinder)

Given, volume of cube=x.
So, length of sides of the cube = \(\sqrt[3]{x}\)
Diagonal of cross-section or Top/Bottom surface(Which is a circle) of the cylinder=Diagonal of cube(Top view is a square-one face of cube)=\(\sqrt{2}*\sqrt[3]{x}\)
or 2* radius=\(\sqrt{2}*\sqrt[3]{x}\)
Or \(r=\frac{\sqrt[3]{x}}{\sqrt{2}}\)

Question stem:- What is the volume of the cylinder?
Vol. of cylinder=\(\pi*r^2*h\)
We need the value of h.

St1:- x=27
So, \(r=\frac{\sqrt[3]{27}}{\sqrt{2}}\)=\(\frac{3}{\sqrt{2}}\)
height is not known or variable. Area of cylinder is not unique.
Insufficient.

St2:- The height of the cylinder is 5 units. Or, h=5 unit
The largest possible length of sides of the cube can be 5 unit . If length of sides of the is greater than 5 unit then it can't be enclosed fully inside the cylinder with h=5 unit.
So, diagonal of circle, 2r=\(5\sqrt{2}\)
Hence, volume of cylinder can be calculated. (radius and height are known)
Sufficient.

Ans. (B)
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The largest possible cube is enclosed in a cylinder and has volume x. 29 Aug 2018, 10:32
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Bunuel
The largest possible cube is enclosed in a cylinder and has volume x. What is the volume of the cylinder?

(1) x is 27.
(2) The height of the cylinder is 5 units.

Refer the top view in the enclosed diagram(When a cube is inserted in a cylinder)

Given, volume of cube=x.
So, length of sides of the cube = \(\sqrt[3]{x}\)
Diagonal of cross-section or Top/Bottom surface(Which is a circle) of the cylinder=Diagonal of cube(Top view is a square-one face of cube)=\(\sqrt{2}*\sqrt[3]{x}\)
or 2* radius=\(\sqrt{2}*\sqrt[3]{x}\)
Or \(r=\frac{\sqrt[3]{x}}{\sqrt{2}}\)

Question stem:- What is the volume of the cylinder?
Vol. of cylinder=\(\pi*r^2*h\)
We need the value of h.

St1:- x=27
So, \(r=\frac{\sqrt[3]{27}}{\sqrt{2}}\)=\(\frac{3}{\sqrt{2}}\)
height is not known or variable. Area of cylinder is not unique.
Insufficient.

St2:- The height of the cylinder is 5 units. Or, h=5 unit
The largest possible length of sides of the cube can be 5 unit . If length of sides of the is greater than 5 unit then it can't be enclosed fully inside the cylinder with h=5 unit.
So, diagonal of circle, 2r=\(5\sqrt{2}\)
Hence, volume of cylinder can be calculated. (radius and height are known)
Sufficient.

Ans. (B)
Show more
Answer is B
( the second statement diagram with front view - - - - > consider height of cylinder to be equal to side of cube---> I have shown height slightly more for the sake of clarity of the diagram)

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1535563942339.jpg [ 61.02 KiB | Viewed 2599 times ]

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The largest possible cube is enclosed in a cylinder and has volume x. 29 Aug 2018, 10:43
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Why, for statement 1, do we not consider that the height of the cube is the height of the cylinder if the question stem tells us that the cube is the maximum possible volume?

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The largest possible cube is enclosed in a cylinder and has volume x. 29 Aug 2018, 10:58
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hastee
Why, for statement 1, do we not consider that the height of the cube is the height of the cylinder if the question stem tells us that the cube is the maximum possible volume?

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Hi hastee,
Can't we put a cube with fixed height into a cylinder with different heights ?(just visualise)

Yes.

In st1 there is no restriction on height of cylinder.

If you have further queries ,you may raise.
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The largest possible cube is enclosed in a cylinder and has volume x. 30 Aug 2018, 02:55
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Bunuel
The largest possible cube is enclosed in a cylinder and has volume x. What is the volume of the cylinder?

(1) x is 27.
(2) The height of the cylinder is 5 units.
Show more


In reference to a PM ...
PKN
Information given about cube
A) largest possible that can fit in cylinder.
B) volume is x ..

Info about cylinder...
Nothing
Required is πr^2h

Statement I : x is 27
So sides of cube are 3..
But we do not know anything about cylinder,
We can just say the height could be 3 , radius could be anything Or
the radius could be 3√2 and height could be anything
Insufficient

Statement II : h is 5
Nothing about the radius
Insufficient

Combined
Height is 5 and not 3, so the restriction on the cube is from radius side.
So radius is 3√2
Volume = π*(3√2)^2*5=90π
Sufficient

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The largest possible cube is enclosed in a cylinder and has volume x. 30 Aug 2018, 03:02
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chetan2u
Bunuel
The largest possible cube is enclosed in a cylinder and has volume x. What is the volume of the cylinder?

(1) x is 27.
(2) The height of the cylinder is 5 units.


In reference to a PM ...
PKN
Information given about cube
A) largest possible that can fit in cylinder.
B) volume is x ..

Info about cylinder...
Nothing
Required is πr^2h

Statement I : x is 27
So sides of cube are 3..
But we do not know anything about cylinder,
We can just say the height could be 3 , radius could be anything Or
the radius could be 3√2 and height could be anything
Insufficient

Statement II : h is 5
Nothing about the radius
Insufficient

Combined
Height is 5 and not 3, so the restriction on the cube is from radius side.
So radius is 3√2
Volume = π*(3√2)^2*5=90π
Sufficient

C
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Thank you Sir.
Could you please point out the fallacies in my reasoning that I have explained in st2.

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 [1]
The largest possible cube is enclosed in a cylinder and has volume x. 30 Aug 2018, 03:17
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PKN
Bunuel
The largest possible cube is enclosed in a cylinder and has volume x. What is the volume of the cylinder?

(1) x is 27.
(2) The height of the cylinder is 5 units.

Refer the top view in the enclosed diagram(When a cube is inserted in a cylinder)

Given, volume of cube=x.
So, length of sides of the cube = \(\sqrt[3]{x}\)
Diagonal of cross-section or Top/Bottom surface(Which is a circle) of the cylinder=Diagonal of cube(Top view is a square-one face of cube)=\(\sqrt{2}*\sqrt[3]{x}\)
or 2* radius=\(\sqrt{2}*\sqrt[3]{x}\)
Or \(r=\frac{\sqrt[3]{x}}{\sqrt{2}}\)

Question stem:- What is the volume of the cylinder?
Vol. of cylinder=\(\pi*r^2*h\)
We need the value of h.

St1:- x=27
So, \(r=\frac{\sqrt[3]{27}}{\sqrt{2}}\)=\(\frac{3}{\sqrt{2}}\)
height is not known or variable. Area of cylinder is not unique.
Insufficient.

St2:- The height of the cylinder is 5 units. Or, h=5 unit
The largest possible length of sides of the cube can be 5 unit . If length of sides of the is greater than 5 unit then it can't be enclosed fully inside the cylinder with h=5 unit.
So, diagonal of circle, 2r=\(5\sqrt{2}\)
Hence, volume of cylinder can be calculated. (radius and height are known)
Sufficient.

Ans. (B)
Show more


Hi..

The flaw is in taking the radius of cylinder dependent on height/on the cube..

It is not 'greatest possible cylinder' that is you could vary the dimensions of cylinder depending on the cube..
But you have a FIXED cylinder, you have to fit in the largest possible cube in it.
So you know now that at least one of the dimensions of cylinder will restrict the cube .
It could be the height or the radius of cylinder..

Height could be 5 but say the radius is just 1, the cube will be different
Height is 3 but radius is 5, now the cube is dependent on height and will be different
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The largest possible cube is enclosed in a cylinder and has volume x. 30 Aug 2018, 03:28
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chetan2u
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Bunuel
The largest possible cube is enclosed in a cylinder and has volume x. What is the volume of the cylinder?

(1) x is 27.
(2) The height of the cylinder is 5 units.

Refer the top view in the enclosed diagram(When a cube is inserted in a cylinder)

Given, volume of cube=x.
So, length of sides of the cube = \(\sqrt[3]{x}\)
Diagonal of cross-section or Top/Bottom surface(Which is a circle) of the cylinder=Diagonal of cube(Top view is a square-one face of cube)=\(\sqrt{2}*\sqrt[3]{x}\)
or 2* radius=\(\sqrt{2}*\sqrt[3]{x}\)
Or \(r=\frac{\sqrt[3]{x}}{\sqrt{2}}\)

Question stem:- What is the volume of the cylinder?
Vol. of cylinder=\(\pi*r^2*h\)
We need the value of h.

St1:- x=27
So, \(r=\frac{\sqrt[3]{27}}{\sqrt{2}}\)=\(\frac{3}{\sqrt{2}}\)
height is not known or variable. Area of cylinder is not unique.
Insufficient.

St2:- The height of the cylinder is 5 units. Or, h=5 unit
The largest possible length of sides of the cube can be 5 unit . If length of sides of the is greater than 5 unit then it can't be enclosed fully inside the cylinder with h=5 unit.
So, diagonal of circle, 2r=\(5\sqrt{2}\)
Hence, volume of cylinder can be calculated. (radius and height are known)
Sufficient.

Ans. (B)


Hi..

The flaw is in taking the radius of cylinder dependent on height/on the cube..

It is not 'greatest possible cylinder' that is you could vary the dimensions of cylinder depending on the cube..
But you have a FIXED cylinder, you have to fit in the largest possible cube in it.
So you know now that at least one of the dimensions of cylinder will restrict the cube .
It could be the height or the radius of cylinder..

Height could be 5 but say the radius is just 1, the cube will be different
Height is 3 but radius is 5, now the cube is dependent on height and will be different
Show more

Thank you again.
I will go through your lines and revert in case of any queries.
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