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

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A sphere is inscribed in a cube with an edge of 10. What is [#permalink]  12 Feb 2012, 20:48
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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) $$5(\sqrt{2} - 1)$$
[Reveal] Spoiler: OA

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Last edited by Bunuel on 12 Feb 2012, 21:21, edited 1 time in total.
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Re: Sphere [#permalink]  12 Feb 2012, 21:20
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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]  13 Feb 2012, 01:24
Wicked Question and great simple explanation Bunuel!
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A sphere is inscribed in a cube with an edge of 10. [#permalink]  17 Jun 2012, 22:03
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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]  18 Jun 2012, 04:47
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]  18 Jun 2012, 04:56
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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 21536 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]  09 Jun 2013, 10:41
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]  09 Jun 2013, 10:54
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Expert's post
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 16112 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]  09 Jun 2013, 11:02
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]  09 Jun 2013, 11:26
Expert's post
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]  09 Jun 2013, 11:32
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]  24 Feb 2014, 01:21
I think what sanjoo meant was not a square in a sphere rather a circle in a square

here it is all three dimensional hence we are considering CUBE and SPHERE

as someone commented above if it were two dimensional like a circle in a square the answer would be D x/2(sqrt(2)-1)
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Re: A sphere is inscribed in a cube with an edge of 10. What is [#permalink]  15 Jun 2014, 11:29
Bunuel wrote:
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
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?

Hi Bunuel,

In your diagram, and in all solutions for questions of this kind, the distance is measured from the diagonal of the cube and not from any other place. Why is this so?
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Re: A sphere is inscribed in a cube with an edge of 10. What is [#permalink]  17 Jun 2014, 21:10
question says "distance from one of the vertices of the cube to the surface of the sphere?".That's why it is calculated from diagnal Even i got similar doubt and even looked for answer Zero immediately. tricky one.
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Re: A sphere is inscribed in a cube with an edge of 10. What is [#permalink]  06 Sep 2014, 10:01
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) $$5(\sqrt{2} - 1)$$

formula-> diagonal of the cube- s*(\sqrt{3}) where s- side of the cube.

sphere inscribed in a cube will be having the diameter of the side of cube. so shortest distance will be (diagonal of the cube - diameter of the sphere) /2 .(when we subtract the diameter from diagonal of the cube we will be left with two side of the cube , hence divided by 2 )

give edge as 10 so sides(cube) as well diameter (sphere) =10.

(10\sqrt{3} - 10 )/2 => option D.
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Re: A sphere is inscribed in a cube with an edge of 10. What is [#permalink]  13 Jan 2015, 22:03
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]  12 Apr 2015, 11:08
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]  12 Apr 2015, 19:04
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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] 12 Apr 2015, 19:04
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