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What is the maximum possible volume of a cube which can be placed insi

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What is the maximum possible volume of a cube which can be placed insi  [#permalink]

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New post 31 Dec 2018, 00:34
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A
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Difficulty:

  55% (hard)

Question Stats:

56% (01:48) correct 44% (01:52) wrong based on 55 sessions

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What is the maximum possible volume of a cube which can be placed inside a sphere with radius 10 units? (in unit^3)

A. 8000/√3
B. 8000/3√3
C. 8000√3
D. 2400√3
E. 1000
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Re: What is the maximum possible volume of a cube which can be placed insi  [#permalink]

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New post 31 Dec 2018, 01:29
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For maximum possible volume of a cube which can be placed inside a sphere, the diameter of the sphere must be equal to the diagonal of the sphere.

Let the radius of sphere be r unit and the side of the cube be a unit.
Given that radius r = 10 units
So, diameter = 2x Radius = 20 units = diagonal of cube
We know diagonal of cube = a√3, where a is the length of each side
Hence, a√3 = 20 units, i. e. a = 20/√3 units
We know volume of cube = a^3 = (20/√3)^3 = 8000/3√3 (unit^3)

Hence, the Correct Answer is Option B. 8000/3√3


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Re: What is the maximum possible volume of a cube which can be placed insi  [#permalink]

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New post 02 Jan 2019, 05:24
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Solution


Given:
    • Radius of the sphere = 10 units

To find:
    • The maximum possible volume of a cube

Approach and Working:
    • Diagonal of the cube, a√3 = diameter of the sphere, 20
      o Implies, \(a = \frac{20}{√3}\)

    • Volume of the cube = \(a^3 = (\frac{20}{√3})^3 = \frac{8000}{3√3}\)

Hence, the correct answer is Option B

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Answer: B
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Re: What is the maximum possible volume of a cube which can be placed insi  [#permalink]

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New post 02 Jan 2019, 18:07
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akurathi12 wrote:
What is the maximum possible volume of a cube which can be placed inside a sphere with radius 10 units? (in unit^3)

A. 8000/√3
B. 8000/3√3
C. 8000√3
D. 2400√3
E. 1000



given radius 10
so diameter = 20

longest side of cube which can fit into sphere
sqrt l^2+b^2+h^2 = 20
since its a cube so l=b=h
or say
sqrt3x^2= 20
squaring both sides
3*x^2=400
or say
x= 20/sqrt3

so vol of cube :
(20/sqrt3)^3 = 8000/3√3

IMO B
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Re: What is the maximum possible volume of a cube which can be placed insi  [#permalink]

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New post 03 Jan 2019, 09:31
DisciplinedPrep wrote:
For maximum possible volume of a cube which can be placed inside a sphere, the diameter of the sphere must be equal to the diagonal of the sphere.

Let the radius of sphere be r unit and the side of the cube be a unit.
Given that radius r = 10 units
So, diameter = 2x Radius = 20 units = diagonal of cube
We know diagonal of cube = a√3, where a is the length of each side
Hence, a√3 = 20 units, i. e. a = 20/√3 units
We know volume of cube = a^3 = (20/√3)^3 = 8000/3√3 (unit^3)

Hence, the Correct Answer is Option B. 8000/3√3



Thanks for the explanation.
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Re: What is the maximum possible volume of a cube which can be placed insi   [#permalink] 03 Jan 2019, 09:31
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