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Math Expert V
Joined: 02 Sep 2009
Posts: 58381
A cube with a volume of 64 cubic inches is inscribed within a sphere  [#permalink]

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Difficulty:   55% (hard)

Question Stats: 66% (01:59) correct 34% (02:15) wrong based on 122 sessions

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A cube with a volume of 64 cubic inches is inscribed within a sphere such that all 8 vertices of the cube are on the sphere. What is the circumference of the sphere, in inches?

A. 2π√2
B. 2π√3
C. 4π√2
D. 4π√3
E. 8π√2

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Re: A cube with a volume of 64 cubic inches is inscribed within a sphere  [#permalink]

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3
1
Bunuel wrote:
A cube with a volume of 64 cubic inches is inscribed within a sphere such that all 8 vertices of the cube are on the sphere. What is the circumference of the sphere, in inches?

A. 2π√2
B. 2π√3
C. 4π√2
D. 4π√3
E. 8π√2

In case of a cube inscribed within a sphere, the cube's diagonal (largest distance between any 2 points on the cube) and the sphere's diameter (largest distance between any 2 points on the sphere) should be equal.
We also know that the diagonal for a cube = $$\sqrt{3}*Side$$

As the volume of the cube = 64 cu. inches, its sides should be 4 inches each.
Thus diagonal = $$4\sqrt{3}$$ = Diameter of the sphere.
Thus radius = $$2\sqrt{3}$$
=> Circumference of the sphere = $$2π*2\sqrt{3}$$ = $$4π√3$$

Should be Option D.
Target Test Prep Representative G
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Re: A cube with a volume of 64 cubic inches is inscribed within a sphere  [#permalink]

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Bunuel wrote:
A cube with a volume of 64 cubic inches is inscribed within a sphere such that all 8 vertices of the cube are on the sphere. What is the circumference of the sphere, in inches?

A. 2π√2
B. 2π√3
C. 4π√2
D. 4π√3
E. 8π√2

Since the cube is inscribed in the sphere, we know that the diagonal of the cube is equal to the diameter of the sphere.

Since the volume of the cube is 64 and volume = side^3, the side is 4. Since the diagonal of a cube is side√3, the diagonal of the cube is 4√3.

Since the sphere’s diameter = 4√3, its radius is 2√3, and thus its circumference = 2πr = 2π(2√3) = 4π√3.

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Re: A cube with a volume of 64 cubic inches is inscribed within a sphere  [#permalink]

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JeffTargetTestPrep wrote:
Bunuel wrote:
A cube with a volume of 64 cubic inches is inscribed within a sphere such that all 8 vertices of the cube are on the sphere. What is the circumference of the sphere, in inches?

A. 2π√2
B. 2π√3
C. 4π√2
D. 4π√3
E. 8π√2

Since the cube is inscribed in the sphere, we know that the diagonal of the cube is equal to the diameter of the sphere.

Since the volume of the cube is 64 and volume = side^3, the side is 4. Since the diagonal of a cube is side√3, the diagonal of the cube is 4√3.

Since the sphere’s diameter = 4√3, its radius is 2√3, and thus its circumference = 2πr = 2π(2√3) = 4π√3.

Certainly so, here is how the figure will look -
Attachment: main-qimg-4fa003c3a8c93fdbba78e63d8f8c4622.png [ 8.09 KiB | Viewed 978 times ]

Answer must be (D)
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Intern  B
Joined: 27 Nov 2017
Posts: 22
Re: A cube with a volume of 64 cubic inches is inscribed within a sphere  [#permalink]

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Hi Bunuel,

Sphere is 3-D object, so will there be circumference for this object or is the question asking about the projection of the sphere? Re: A cube with a volume of 64 cubic inches is inscribed within a sphere   [#permalink] 28 Oct 2018, 00:54
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