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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]

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New post 10 Sep 2010, 06:39
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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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A sphere is inscribed in a cube with an edge of 10. What is  [#permalink]

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New post 10 Sep 2010, 10:41
5
4
TehJay wrote:
How did you come up with that formula for the shortest distance from the vertex to the sphere?


It would be easier if you visualize this problem.

As sphere is inscribed in cube then the edge of the cube equals 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-Diameter is gap between the vertex of a cube and the surface of the sphere:

\(x=\frac{Diagonal -Diameter}{2}=\frac{10*\sqrt{3}-10}{2}=5(\sqrt{3}-1)\)
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Re: Cube + sphere  [#permalink]

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New post 10 Sep 2010, 06:51
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rxs0005 wrote:
A sphere is inscribed in a cube with edge 10. The sphere touches the edges of the cube

What is the shortest distance from any vertex of the cube to the surface of the sphere

5

10

5*root(2) -1


10*root(2) - 1

2*root(5) - 1


Please read and follow: how-to-improve-the-forum-search-function-for-others-99451.html

Original question, with correct answer choices and correct OA is as follows:

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) \(5(\sqrt{2} - 1)\)

Shortest distance=(diagonal of cube-diameter of sphere)/2= \(\frac{10*\sqrt{3}-10}{2}=5(\sqrt{3}-1)\)

Answer: D.
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Re: Cube + sphere  [#permalink]

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New post 10 Sep 2010, 10:20
How did you come up with that formula for the shortest distance from the vertex to the sphere?
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Re: Cube + sphere  [#permalink]

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New post 10 Sep 2010, 10:48
Bunuel wrote:
TehJay wrote:
How did you come up with that formula for the shortest distance from the vertex to the sphere?


It would be easier if you visualize this problem.

As sphere is inscribed in cube then the edge of the cube equals 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-Diameter is gap between the vertex of a cube and the surface of the sphere --> \(x=\frac{Diagonal -Diameter}{2}=\frac{10*\sqrt{3}-10}{2}=5(\sqrt{3}-1)\)


Thanks, I get it now!
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A sphere is inscribed in a cube with an edge of 10  [#permalink]

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New post 14 Dec 2010, 06:32
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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?
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Re: A sphere is inscribed in a cube with an edge of 10. What is  [#permalink]

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New post 16 Oct 2013, 01:48
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) \(5(\sqrt{2} - 1)\)


Similar questions to practice:
a-sphere-is-inscribed-in-a-cube-with-an-edge-of-10-what-is-127461.html
a-rectangular-box-has-dimensions-of-8-feet-8-feet-and-z-128483.html
for-the-cube-shown-above-what-is-the-degree-measure-of-pqr-13841.html
a-rectangular-box-is-10-inches-wide-10-inches-long-and-144733.html
if-the-box-shown-is-a-cube-then-the-difference-in-length-127463.html
what-is-the-volume-of-the-cube-above-103680.html
if-a-cube-is-inscribed-inside-a-sphere-154770.html

Hope it helps.
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Collection of Questions:
PS: 1. Tough and Tricky questions; 2. Hard questions; 3. Hard questions part 2; 4. Standard deviation; 5. Tough Problem Solving Questions With Solutions; 6. Probability and Combinations Questions With Solutions; 7 Tough and tricky exponents and roots questions; 8 12 Easy Pieces (or not?); 9 Bakers' Dozen; 10 Algebra set. ,11 Mixed Questions, 12 Fresh Meat

DS: 1. DS tough questions; 2. DS tough questions part 2; 3. DS tough questions part 3; 4. DS Standard deviation; 5. Inequalities; 6. 700+ GMAT Data Sufficiency Questions With Explanations; 7 Tough and tricky exponents and roots questions; 8 The Discreet Charm of the DS; 9 Devil's Dozen!!!; 10 Number Properties set., 11 New DS set.


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Re: Cube + sphere  [#permalink]

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New post 08 Feb 2014, 02:53
Bunuel wrote:
TehJay wrote:
How did you come up with that formula for the shortest distance from the vertex to the sphere?


It would be easier if you visualize this problem.

As sphere is inscribed in cube then the edge of the cube equals 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-Diameter is gap between the vertex of a cube and the surface of the sphere --> \(x=\frac{Diagonal -Diameter}{2}=\frac{10*\sqrt{3}-10}{2}=5(\sqrt{3}-1)\)



What i didn't comprehend was why 1/2. How can we evaluate that it is 1/2 ? is it a formula that we have to cram?
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Re: Cube + sphere  [#permalink]

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New post 08 Feb 2014, 05:19
Abheek wrote:
Bunuel wrote:
TehJay wrote:
How did you come up with that formula for the shortest distance from the vertex to the sphere?


It would be easier if you visualize this problem.

As sphere is inscribed in cube then the edge of the cube equals 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-Diameter is gap between the vertex of a cube and the surface of the sphere --> \(x=\frac{Diagonal -Diameter}{2}=\frac{10*\sqrt{3}-10}{2}=5(\sqrt{3}-1)\)



What i didn't comprehend was why 1/2. How can we evaluate that it is 1/2 ? is it a formula that we have to cram?


It would be easier if you visualize. Check the diagrams here: a-sphere-is-inscribed-in-a-cube-with-an-edge-of-10-what-is-127461.html and here: a-sphere-is-inscribed-in-a-cube-with-an-edge-of-10-what-is-127461.html#p1234095

Hope it helps.
_________________

New to the Math Forum?
Please read this: Ultimate GMAT Quantitative Megathread | All You Need for Quant | PLEASE READ AND FOLLOW: 12 Rules for Posting!!!

Resources:
GMAT Math Book | Triangles | Polygons | Coordinate Geometry | Factorials | Circles | Number Theory | Remainders; 8. Overlapping Sets | PDF of Math Book; 10. Remainders | GMAT Prep Software Analysis | SEVEN SAMURAI OF 2012 (BEST DISCUSSIONS) | Tricky questions from previous years.

Collection of Questions:
PS: 1. Tough and Tricky questions; 2. Hard questions; 3. Hard questions part 2; 4. Standard deviation; 5. Tough Problem Solving Questions With Solutions; 6. Probability and Combinations Questions With Solutions; 7 Tough and tricky exponents and roots questions; 8 12 Easy Pieces (or not?); 9 Bakers' Dozen; 10 Algebra set. ,11 Mixed Questions, 12 Fresh Meat

DS: 1. DS tough questions; 2. DS tough questions part 2; 3. DS tough questions part 3; 4. DS Standard deviation; 5. Inequalities; 6. 700+ GMAT Data Sufficiency Questions With Explanations; 7 Tough and tricky exponents and roots questions; 8 The Discreet Charm of the DS; 9 Devil's Dozen!!!; 10 Number Properties set., 11 New DS set.


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

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New post 20 Jan 2015, 15:30
Can someone let me know what is wrong with my approach please.

1. I take the corner of the square to the centre of the square
2. Then subtract out the radius of the circle (all in 2dimension)
3. Use 3d Pythagorus to get distance to the sphere

I do not arrive at the correct answer and instead get 5(rt(3)*rt(2)-1)

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

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New post 09 Aug 2015, 09:42
Why can you not think of the problem as a pyramid formed by a vertex of the cube, the closest point where the cube's diagonal intersects the sphere, and an adjacent tangent point between the cube and sphere? Then apply 45/45/90 logic? It doesn't seem to yield the same answer, but IDK where I went wrong. Thx!
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Re: A sphere is inscribed in a cube with an edge of 10. What is  [#permalink]

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New post 09 Apr 2018, 08:06
bigfoot wrote:
Can someone let me know what is wrong with my approach please.

1. I take the corner of the square to the centre of the square
2. Then subtract out the radius of the circle (all in 2dimension)
3. Use 3d Pythagorus to get distance to the sphere

I do not arrive at the correct answer and instead get 5(rt(3)*rt(2)-1)

Thanks


Hey

Can u upload a photo of your calculations.. I went by a similar method and got the right answer.
Though you can refer the problems highlighted by Bunuel
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Re: A sphere is inscribed in a cube with an edge of 10. What is &nbs [#permalink] 09 Apr 2018, 08:06
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