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Math Expert V
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A sphere, diameter 6, is cut to form a cube. What is the maximum possi  [#permalink]

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Difficulty:   45% (medium)

Question Stats: 63% (01:25) correct 38% (02:12) wrong based on 80 sessions

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Competition Mode Question

A sphere, diameter 6, is cut to form a cube. What is the maximum possible distance between two vertices after the cut?

A. $$3$$

B. $$3\sqrt{2}$$

C. $$2\pi$$

D. $$6$$

E. $$3(\pi-2)$$

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Re: A sphere, diameter 6, is cut to form a cube. What is the maximum possi  [#permalink]

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We can look at this question as a square inscribed within a circle. Since the edges of the square will touch the circle, then the diagonal is equal to the diameter of the circle. And by this extension, the maximum possible distance between any two edges if a cube is cut out of a sphere will be equal to the diameter of the sphere.

Since we know that the diameter of the sphere is 6 units, then the maximum distance between the vertices will be 6 units as well.

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Re: A sphere, diameter 6, is cut to form a cube. What is the maximum possi  [#permalink]

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A sphere, diameter 6, is cut to form a cube. What is the maximum possible distance between two vertices after the cut?

Since a cube with side ‘a’ has longest distance equal to √3a, all the vertices of the cube cut out from a sphere of diameter 6 would touch the sphere’s surface such that cube is of largest size.

Hence √3a = 6.

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Re: A sphere, diameter 6, is cut to form a cube. What is the maximum possi  [#permalink]

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digonal of cube= diameter of circle ; 6 will be the maximum distance b/w two vertices
side ; 3√2 each of the cube
IMO D

A sphere, diameter 6, is cut to form a cube. What is the maximum possible distance between two vertices after the cut?

A. 33

B. 32‾√32

C. 2π2π

D. 66

E. 3(π−2)
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Re: A sphere, diameter 6, is cut to form a cube. What is the maximum possi  [#permalink]

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Quote:
A sphere, diameter 6, is cut to form a cube. What is the maximum possible distance between two vertices after the cut?

A. 3
B. 2√3
C. 2π
D. 6
E. 3(π−2)

max distance between two vertices of the largest cube inscribed in a sphere is:
cube's long diagonal = spheres diameter = 6

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Re: A sphere, diameter 6, is cut to form a cube. What is the maximum possi  [#permalink]

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Since the sphere is cut to form the cube.

The maximum possible cube would be the one which is can be fitted inside the cube.

Hence the line connecting the opposite vertices will the maximum possible distance between any two vertex, which will be same as that of diameter of the sphere.

Hence ans -6 (D)
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Re: A sphere, diameter 6, is cut to form a cube. What is the maximum possi  [#permalink]

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Maximum distance between two vertices must be diameter of the sphere which is 6.

IMO D
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Joined: 11 Aug 2019
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Re: A sphere, diameter 6, is cut to form a cube. What is the maximum possi  [#permalink]

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Bunuel wrote:

Competition Mode Question

A sphere, diameter 6, is cut to form a cube. What is the maximum possible distance between two vertices after the cut?

A. $$3$$

B. $$3\sqrt{2}$$

C. $$2\pi$$

D. $$6$$

E. $$3(\pi-2)$$

Bunuel Please explain where I got this wrong.

Imagine cutting the sphere into two halves and we see the cube (cut in half) inside.
Since the diameter is 6, AB should be 6.
Then we can find the sides of the square, which is 3 * sqr2 each.
So the cube's dimension is: 3 * sqr2

It's diagonal is: 3 * sqr2 * sqr 3.

The distance is therefore: 3 * Sqr6.

Thanks!
Attachments GMAT circle question.png [ 12.27 KiB | Viewed 402 times ]

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Re: A sphere, diameter 6, is cut to form a cube. What is the maximum possi  [#permalink]

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Bunuel wrote:

Competition Mode Question

A sphere, diameter 6, is cut to form a cube. What is the maximum possible distance between two vertices after the cut?

A. $$3$$

B. $$3\sqrt{2}$$

C. $$2\pi$$

D. $$6$$

E. $$3(\pi-2)$$

If a sphere is cut to form a cube, the largest cube that can be formed is the cube that can be inscribed in the sphere. Therefore, the maximum distance between two vertices of the cube is also the length of the diameter of the sphere, which is 6.

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