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A sphere is inscribed in a cube with an edge of 10. What is  [#permalink]

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Question Stats: 55% (01:39) correct 45% (01:49) wrong based on 804 sessions

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A sphere is inscribed in a cube with an edge of 10. What is the shortest possible distance from one of the vertices of the cube to the surface of the sphere?

(A) $$10(\sqrt{3}- 1)$$
(B) $$5$$
(C) $$10(\sqrt{2} - 1)$$
(D) $$5(\sqrt{3} - 1)$$
(E) $$5(\sqrt{2} - 1)$$

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Originally posted by enigma123 on 12 Feb 2012, 20:48.
Last edited by Bunuel on 12 Feb 2012, 21:21, edited 1 time in total.
Edited the question
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A sphere is inscribed in a cube with an edge of 10. What is the shortest possible distance from one of the vertices of the cube to the surface of the sphere?
(A) $$10(\sqrt{3}- 1)$$
(B) $$5$$
(C) $$10(\sqrt{2} - 1)$$
(D) $$5(\sqrt{3} - 1)$$
(E) $$5(\sqrt{2} - 1)$$

It would be easier if you visualize this problem.

As sphere is inscribed in cube then the edges of the cube equal to the diameter of a sphere --> $$Diameter=10$$.

Next, diagonal of a cube equals to $$Diagonal=\sqrt{10^2+10^2+10^2}=10\sqrt{3}$$.

Now half of (Diagonal minus Diameter) is a gap between the vertex of a cube and the surface of the sphere, which will be the shortest distance: $$x=\frac{Diagonal -Diameter}{2}=\frac{10*\sqrt{3}-10}{2}=5(\sqrt{3}-1)$$

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Re: A sphere is inscribed in a cube with an edge of 10. What is  [#permalink]

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Wicked Question and great simple explanation Bunuel!
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A sphere is inscribed in a cube with an edge of 10.  [#permalink]

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Also if it were not sphere and just a two dimensional square, the shortest distance would be S/2(\sqrt{2} - 1) = 10/2(\sqrt{2} - 1) = 5(\sqrt{2} - 1)
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Re: A sphere is inscribed in a cube with an edge of 10. What is  [#permalink]

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Bunuel.. i cant understand the question ? can u elaborate it further..

And ya instead of sphere if it wud be a square inscribed in a cube then wat wud b the answer?
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Re: A sphere is inscribed in a cube with an edge of 10. What is  [#permalink]

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sanjoo wrote:
Bunuel.. i cant understand the question ? can u elaborate it further..

And ya instead of sphere if it wud be a square inscribed in a cube then wat wud b the answer?

Look at the diagram below:
Attachment: Sphere inscribed in a cube.png [ 4.02 KiB | Viewed 105983 times ]
The question asks about the lengths of the little black arrows shown.

As for the additional question, it doesn't make much sense: what does it mean a square is inscribed in a cube?
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Re: A sphere is inscribed in a cube with an edge of 10. What is  [#permalink]

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i dint understood why diameter is 10?
how smallest area is 1/2(diameter-diagonal) bunuel help
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Re: A sphere is inscribed in a cube with an edge of 10. What is  [#permalink]

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WarriorGmat wrote:
i dint understood why diameter is 10?
how smallest area is 1/2(diameter-diagonal) bunuel help

Consider the cross-section as shown below:
Attachment: square.png [ 3.86 KiB | Viewed 85475 times ]
The diameter = The edge.

As for your second question, check here: a-sphere-is-inscribed-in-a-cube-with-an-edge-of-10-what-is-127461.html#p1097531 The shortest distance from one of the vertices of the cube to the surface of the sphere is 1/2(diagonal of the cube - diameter of the circle). Diagonal of the cube - diameter of the circle, is the length of two little black arrows shown here: Hope it's clear.
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Re: A sphere is inscribed in a cube with an edge of 10. What is  [#permalink]

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hi bunuel
thanks for an explanation.
why diagonal-diameter divided by 2
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Re: A sphere is inscribed in a cube with an edge of 10. What is  [#permalink]

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WarriorGmat wrote:
hi bunuel
thanks for an explanation.
why diagonal-diameter divided by 2

Diagonal - diameter is the length of two little black arrows we need one...
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Re: A sphere is inscribed in a cube with an edge of 10. What is  [#permalink]

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thanks bunuel for patiently providin g solution
you rock!!
too much and too little study is fatal thats what happening to me i have missed such a small stuff.

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Re: A sphere is inscribed in a cube with an edge of 10. What is  [#permalink]

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rxs0005 wrote:
A sphere is inscribed in a cube with an edge of 10. What is the shortest possible distance from one of the vertices of the cube to the surface of the sphere?

(A) $$10(\sqrt{3}- 1)$$
(B) $$5$$
(C) $$10(\sqrt{2} - 1)$$
(D) $$5(\sqrt{3} - 1)$$
(E) $$5(\sqrt{2} - 1)$$

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Hope it helps.
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Re: A sphere is inscribed in a cube with an edge of 10. What is  [#permalink]

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If you visualize the problem in your head, you realize that what you want is 1/2 the diagonal of the cube- the radius of the circle.

We know the radius of the circle is 5, because the circle touches the sided of the cube, which has a total length of 10.

If you memorized the diagonal of a cube, which I found helpful to do for my test, then you would know it is side*(sqrt(3)), but we want half of that so it is 5(sqrt(3)).

So, the distance of the vertice to the sphere is 5(sqrt(3))-5. That is not an answer, but we can see that D is the same thing, it just divided out the 5.
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Re: A sphere is inscribed in a cube with an edge of 10. What is  [#permalink]

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I did not get how the diameter is 100 in this step
distance: x=Diagonal−Diameter2=10∗√3−102=5(√3−1)

Could you plz explain?
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Re: A sphere is inscribed in a cube with an edge of 10. What is  [#permalink]

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Hi SonaliT,

Since the sphere is inscribed in the cube, it's diameter is the SAME as the edge of the cube. They are BOTH 10 (not 100).

The calculation that you referred to should be written as....

10(Root 3) - 10 = 10(Root 3 -1)

This calculation is TWICE the length that we're looking for (one on both "sides" of the sphere). Since the question asks for the shortest distance from any of the vertices on the cube to the sphere, we have to divide this entire calculation by 2....

10(Root 3 - 1)/2 = 5(Root 3 - 1)

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Re: A sphere is inscribed in a cube with an edge of 10. What is  [#permalink]

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Here is the official explanation.
Attachments cubeSphere.jpg [ 94.97 KiB | Viewed 59507 times ]

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Re: A sphere is inscribed in a cube with an edge of 10. What is  [#permalink]

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enigma123 wrote:
A sphere is inscribed in a cube with an edge of 10. What is the shortest possible distance from one of the vertices of the cube to the surface of the sphere?

(A) $$10(\sqrt{3}- 1)$$
(B) $$5$$
(C) $$10(\sqrt{2} - 1)$$
(D) $$5(\sqrt{3} - 1)$$
(E) $$5(\sqrt{2} - 1)$$

Answer: Option D (Just replace edge length as 10 instead of 20)

Check solution as attached
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_________________ Re: A sphere is inscribed in a cube with an edge of 10. What is   [#permalink] 11 May 2020, 00:41

# A sphere is inscribed in a cube with an edge of 10. What is  