Check GMAT Club Decision Tracker for the Latest School Decision Releases https://gmatclub.com/AppTrack
GMAT Club

 It is currently 27 Mar 2017, 19:35

### GMAT Club Daily Prep

#### Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email.

Customized
for You

we will pick new questions that match your level based on your Timer History

Track

every week, we’ll send you an estimated GMAT score based on your performance

Practice
Pays

we will pick new questions that match your level based on your Timer History

# Events & Promotions

###### Events & Promotions in June
Open Detailed Calendar

# A sphere is inscribed in a cube with an edge of x centimeter

Author Message
TAGS:

### Hide Tags

Senior Manager
Joined: 02 Sep 2012
Posts: 259
Location: United States
Concentration: Entrepreneurship, Finance
GMAT Date: 07-25-2013
GPA: 3.83
WE: Architecture (Computer Hardware)
Followers: 5

Kudos [?]: 187 [0], given: 99

A sphere is inscribed in a cube with an edge of x centimeter [#permalink]

### Show Tags

20 Jul 2013, 05:03
5
This post was
BOOKMARKED
00:00

Difficulty:

45% (medium)

Question Stats:

57% (01:42) correct 43% (00:55) wrong based on 225 sessions

### HideShow timer Statistics

A sphere is inscribed in a cube with an edge of x centimeters. In terms of x what is the shortest possible distance from one of the vertices of the cube to the surface of the sphere?

A. $$x(\sqrt{3}- 1)$$
B. $$\frac{x}{2}$$
C. $$x(\sqrt{2} - 1)$$
D. $$\frac{x}{2}(\sqrt{3} - 1)$$
E. $$\frac{x}{2}(\sqrt{2} - 1)$$

M28-01
[Reveal] Spoiler: OA

_________________

"Giving kudos" is a decent way to say "Thanks" and motivate contributors. Please use them, it won't cost you anything

Last edited by Bunuel on 20 Jul 2013, 06:17, edited 1 time in total.
Edited the question.
Math Expert
Joined: 02 Sep 2009
Posts: 37619
Followers: 7406

Kudos [?]: 99716 [3] , given: 11035

Re: A sphere is inscribed in a cube with an edge of x centimeter [#permalink]

### Show Tags

20 Jul 2013, 06:18
3
KUDOS
Expert's post
2
This post was
BOOKMARKED
skamal7 wrote:
A sphere is inscribed in a cube with an edge of x centimeters. In terms of x what is the shortest possible distance from one of the vertices of the cube to the surface of the sphere?

A. $$x(\sqrt{3}- 1)$$
B. $$\frac{x}{2}$$
C. $$x(\sqrt{2} - 1)$$
D. $$\frac{x}{2}(\sqrt{3} - 1)$$
E. $$\frac{x}{2}(\sqrt{2} - 1)$$

M28-01

Say $$x=10$$ centimeters.

Then, since a 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 minus Diameter is gap between the vertex of a cube and the surface of the sphere --> $$gap=\frac{Diagonal -Diameter}{2}=\frac{10*\sqrt{3}-10}{2}=5(\sqrt{3}-1)$$.

Since $$x=10$$ then $$5(\sqrt{3}-1)=\frac{x}{2}(\sqrt{3} - 1)$$.

Similar question to practice: a-sphere-is-inscribed-in-a-cube-with-an-edge-of-10-what-is-127461.html
_________________
Senior Manager
Joined: 02 Sep 2012
Posts: 259
Location: United States
Concentration: Entrepreneurship, Finance
GMAT Date: 07-25-2013
GPA: 3.83
WE: Architecture (Computer Hardware)
Followers: 5

Kudos [?]: 187 [0], given: 99

Re: A sphere is inscribed in a cube with an edge of x centimeter [#permalink]

### Show Tags

20 Jul 2013, 06:48
Wow awesome explanation !! +1 to you..By any chance are there any questions which are similar to this other apart from the link provided by you?
_________________

"Giving kudos" is a decent way to say "Thanks" and motivate contributors. Please use them, it won't cost you anything

Intern
Joined: 29 Oct 2013
Posts: 3
Location: United States
Concentration: Technology, General Management
Schools: McCombs '16 (WL)
GMAT 1: 670 Q V0
GPA: 3.6
Followers: 0

Kudos [?]: 5 [0], given: 0

Re: A sphere is inscribed in a cube with an edge of x centimeter [#permalink]

### Show Tags

12 Nov 2013, 14:36
Why do we have to go along the diagonal of the cube?

I am getting E by going straight from the corner towards the sphere. The way I'm doing it would be the same if it were a circle/square rather than sphere/cube.

Why is this wrong? I can't figure it out
Veritas Prep GMAT Instructor
Joined: 16 Oct 2010
Posts: 7252
Location: Pune, India
Followers: 2205

Kudos [?]: 14355 [7] , given: 222

Re: A sphere is inscribed in a cube with an edge of x centimeter [#permalink]

### Show Tags

13 Nov 2013, 00:44
7
KUDOS
Expert's post
1
This post was
BOOKMARKED
gmatzac wrote:
Why do we have to go along the diagonal of the cube?

I am getting E by going straight from the corner towards the sphere. The way I'm doing it would be the same if it were a circle/square rather than sphere/cube.

Why is this wrong? I can't figure it out

That's not correct. Draw a cube and see how you will inscribe a sphere in it. Note that the sphere will not touch any edges/corners of the cube. It will touch only the 6 faces of the cube at one point each. This point will lie in the center of the face of the cube.
If you go the usual two dimensional way, you are assuming that the sphere is lying flat on the face of the cube which is not correct. The sphere only touches the face of the cube on one point i.e. the point where the diagonals of the square face intersect. Hence, actually the distance of this diagonal to the sphere will be half the length of the diagonal. On the other hand, the diagonal of the cube (from one vertex to the opposite vertex across the cube will go right through the center of the sphere. It will stick a little bit out on both sides close to the vertex but will predominantly lie within the sphere on its diameter. So we find the length of the cube diagonal, subtract the sphere diameter out of it and divide the rest of the diagonal by 2 to get length of each little piece.

Think of a globe and its inclined axis. Imagine making a cube around it such that the globe touches each face of the cube. The shortest distance between a vertex of the cube and the globe will be the part of the inclined axis sticking out of the globe touching a vertex of the cube.
_________________

Karishma
Veritas Prep | GMAT Instructor
My Blog

Get started with Veritas Prep GMAT On Demand for $199 Veritas Prep Reviews Intern Joined: 29 Oct 2013 Posts: 3 Location: United States Concentration: Technology, General Management Schools: McCombs '16 (WL) GMAT 1: 670 Q V0 GPA: 3.6 Followers: 0 Kudos [?]: 5 [0], given: 0 Re: A sphere is inscribed in a cube with an edge of x centimeter [#permalink] ### Show Tags 13 Nov 2013, 12:37 Wow! You made the visualization perfectly clear. I was thinking cylinder not sphere, however your explanation made the second half of solving the problem make the algebra come together easily. Thanks so much for that reply, I really appreciate it. GMAT Club Legend Joined: 09 Sep 2013 Posts: 14478 Followers: 609 Kudos [?]: 174 [0], given: 0 Re: A sphere is inscribed in a cube with an edge of x centimeter [#permalink] ### Show Tags 18 Feb 2015, 21:18 Hello from the GMAT Club BumpBot! Thanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up - doing my job. I think you may find it valuable (esp those replies with Kudos). Want to see all other topics I dig out? Follow me (click follow button on profile). You will receive a summary of all topics I bump in your profile area as well as via email. _________________ Senior Manager Status: Math is psycho-logical Joined: 07 Apr 2014 Posts: 442 Location: Netherlands GMAT Date: 02-11-2015 WE: Psychology and Counseling (Other) Followers: 2 Kudos [?]: 114 [0], given: 169 A sphere is inscribed in a cube with an edge of x centimeter [#permalink] ### Show Tags 25 Feb 2015, 14:46 I did it using the relationship between the diagonal of the cube and the side of the cube. So, if the side is x, the diagonal is xSQRT3. So, to find the gap, we need to subtract the diameter of the circle from the diagonal of the square (which would leave us with the two small gaps between the circle and the square, across the diagonal of he square) and divide this by 2, to get only the length of one of the gaps. So, the diagonal is xSQRT3 The diameter of the circle is x, as the side of the square ( this is obvious if you draw the diagonal in the middle of the square, where the circle is touching the sides of the square). (xSQRT3 - x) / 2 = x/2 (SQRT3 - 1). Sorry for the ugly formatting, but I couldn't do the square roots in preview... Verbal Forum Moderator Joined: 29 Apr 2015 Posts: 899 Location: Switzerland Concentration: Economics, Finance Schools: LBS MIF '19 WE: Asset Management (Investment Banking) Followers: 53 Kudos [?]: 1337 [0], given: 302 Re: A sphere is inscribed in a cube with an edge of x centimeter [#permalink] ### Show Tags 29 Aug 2015, 05:11 Bunuel wrote: Now half of Diagonal minus Diameter is gap between the vertex of a cube and the surface of the sphere Does GMAC require this knowledge about spheres and these distances? _________________ Saving was yesterday, heat up the gmatclub.forum's sentiment by spending KUDOS! PS Please send me PM if I do not respond to your question within 24 hours. Veritas Prep GMAT Instructor Joined: 16 Oct 2010 Posts: 7252 Location: Pune, India Followers: 2205 Kudos [?]: 14355 [1] , given: 222 Re: A sphere is inscribed in a cube with an edge of x centimeter [#permalink] ### Show Tags 31 Aug 2015, 09:07 1 This post received KUDOS Expert's post reto wrote: Bunuel wrote: Now half of Diagonal minus Diameter is gap between the vertex of a cube and the surface of the sphere Does GMAC require this knowledge about spheres and these distances? This question requires no particular knowledge about spheres. It needs you to just visualise - nothing wrong with that. It is certainly suitable for GMAT. You could get a volume of sphere kind of question too but you will be given the formula used to find the volume of sphere. For a regular 3-D figure such as a cylinder or prism (where Volume = Area of base * Height), you could be required to find the volume without the formula. _________________ Karishma Veritas Prep | GMAT Instructor My Blog Get started with Veritas Prep GMAT On Demand for$199

Veritas Prep Reviews

Veritas Prep GMAT Instructor
Joined: 16 Oct 2010
Posts: 7252
Location: Pune, India
Followers: 2205

Kudos [?]: 14355 [2] , given: 222

Re: A sphere is inscribed in a cube with an edge of x centimeter [#permalink]

### Show Tags

02 Nov 2015, 23:07
2
KUDOS
Expert's post
VeritasPrepKarishma wrote:
gmatzac wrote:
Why do we have to go along the diagonal of the cube?

I am getting E by going straight from the corner towards the sphere. The way I'm doing it would be the same if it were a circle/square rather than sphere/cube.

Why is this wrong? I can't figure it out

That's not correct. Draw a cube and see how you will inscribe a sphere in it. Note that the sphere will not touch any edges/corners of the cube. It will touch only the 6 faces of the cube at one point each. This point will lie in the center of the face of the cube.
If you go the usual two dimensional way, you are assuming that the sphere is lying flat on the face of the cube which is not correct. The sphere only touches the face of the cube on one point i.e. the point where the diagonals of the square face intersect. Hence, actually the distance of this diagonal to the sphere will be half the length of the diagonal. On the other hand, the diagonal of the cube (from one vertex to the opposite vertex across the cube will go right through the center of the sphere. It will stick a little bit out on both sides close to the vertex but will predominantly lie within the sphere on its diameter. So we find the length of the cube diagonal, subtract the sphere diameter out of it and divide the rest of the diagonal by 2 to get length of each little piece.

Think of a globe and its inclined axis. Imagine making a cube around it such that the globe touches each face of the cube. The shortest distance between a vertex of the cube and the globe will be the part of the inclined axis sticking out of the globe touching a vertex of the cube.

Responding to a pm:

Quote:
CAN YOU please explain why is the diagonal root-square 10^2*10^2*10^2 and not just 10^2*10^2 (applying P.theor.)?

Diagonal of a square will be $$\sqrt{(10^2 + 10^2)}$$ (shown by 'd' in the diagram)
Diagonal of a cube will be 3 dimensional (shown by 'D' in the figure - the green line). We will need to use pythagorean theorem again on it. It will be the hypotenuse when the legs are height of the cube (a) and the diagonal of the square face (d).
Attachment:

cube111.PNG [ 3.19 KiB | Viewed 2573 times ]

$$D = \sqrt{d^2 + a^2} = \sqrt{{sqrt(10^2 + 10^2)}^2 + 10^2}} = \sqrt{10^2 + 10^2 + 10^2}$$
_________________

Karishma
Veritas Prep | GMAT Instructor
My Blog

Get started with Veritas Prep GMAT On Demand for \$199

Veritas Prep Reviews

Math Expert
Joined: 02 Sep 2009
Posts: 37619
Followers: 7406

Kudos [?]: 99716 [0], given: 11035

Re: A sphere is inscribed in a cube with an edge of x centimeter [#permalink]

### Show Tags

03 Nov 2015, 02:06
Expert's post
1
This post was
BOOKMARKED
skamal7 wrote:
A sphere is inscribed in a cube with an edge of x centimeters. In terms of x what is the shortest possible distance from one of the vertices of the cube to the surface of the sphere?

A. $$x(\sqrt{3}- 1)$$
B. $$\frac{x}{2}$$
C. $$x(\sqrt{2} - 1)$$
D. $$\frac{x}{2}(\sqrt{3} - 1)$$
E. $$\frac{x}{2}(\sqrt{2} - 1)$$

M28-01

Similar question to practice:
a-carpenter-wants-to-ship-a-copper-rod-to-his-new-construction-locatio-187870.html
a-rectangular-box-is-12-inches-wide-16-inches-long-and-20-inches-hig-201511.html
the-figure-above-is-a-cube-what-is-the-measure-of-angle-beg-126246.html
for-the-cube-shown-above-what-is-the-degree-measure-of-pqr-13841.html
the-figure-above-is-a-cube-what-is-the-measure-of-angle-beg-129650.html
square-abcd-is-the-base-of-the-cube-while-square-efgh-is-the-56270.html
a-cube-has-sides-measuring-6-inches-what-is-the-greatest-153098.html
a-sphere-is-inscribed-in-a-cube-with-an-edge-of-10-what-is-127461.html
a-sphere-is-inscribed-in-a-cube-with-an-edge-of-10-what-is-100798.html
if-the-box-shown-is-a-cube-then-the-difference-in-length-be-82136.html
a-rectangular-box-has-dimensions-of-8-feet-8-feet-and-z-128483.html
if-the-box-shown-is-a-cube-then-the-difference-in-length-127463.html
a-rectangular-box-is-10-inches-wide-10-inches-long-and-144733.html
a-rectangular-box-has-dimensions-of-8-feet-8-feet-and-z-128483.html
q-is-a-cube-find-the-volume-of-the-cube-193709.html
what-is-the-volume-of-the-cube-above-103680.html
if-a-cube-is-inscribed-inside-a-sphere-154770.html
what-is-the-volume-of-a-certain-cube-164377.html
_________________
Director
Joined: 05 Mar 2015
Posts: 808
Followers: 9

Kudos [?]: 154 [0], given: 39

A sphere is inscribed in a cube with an edge of x centimeter [#permalink]

### Show Tags

17 Jul 2016, 11:58
skamal7 wrote:
A sphere is inscribed in a cube with an edge of x centimeters. In terms of x what is the shortest possible distance from one of the vertices of the cube to the surface of the sphere?

A. $$x(\sqrt{3}- 1)$$
B. $$\frac{x}{2}$$
C. $$x(\sqrt{2} - 1)$$
D. $$\frac{x}{2}(\sqrt{3} - 1)$$
E. $$\frac{x}{2}(\sqrt{2} - 1)$$

M28-01

Attachment:

look at the fig.
diagonal of cube=[square_root]3*x
but sphere of radius x is in between cube so left out distance from two opposite vertices =[square_root]3*x-x
but we need only any one side distance --->1/2([square_root]3*x-x)
x/2([square_root]3*-1)
Ans D
A sphere is inscribed in a cube with an edge of x centimeter   [#permalink] 17 Jul 2016, 11:58
Similar topics Replies Last post
Similar
Topics:
2 If the length of an edge of cube X is twice the length of an 3 03 Aug 2013, 20:20
95 A sphere is inscribed in a cube with an edge of 10. What is 20 12 Feb 2012, 21:48
1 A sphere is inscribed in a cube with an edge of 10. What is 2 22 Mar 2011, 14:36
25 A sphere is inscribed in a cube with an edge of 10. What is 13 10 Sep 2010, 07:39
9 Sphere is inscribed in a cube of side length 10 cms. What is 3 30 Jun 2007, 18:52
Display posts from previous: Sort by