GMAT Question of the Day - Daily to your Mailbox; hard ones only

 It is currently 13 Dec 2019, 05:29 ### 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

#### Not interested in getting valuable practice questions and articles delivered to your email? No problem, unsubscribe here.  # A sphere is inscribed in a cube with an edge of 10. What is

Author Message
TAGS:

### Hide Tags

Senior Manager  Status: Finally Done. Admitted in Kellogg for 2015 intake
Joined: 25 Jun 2011
Posts: 439
Location: United Kingdom
GMAT 1: 730 Q49 V45 GPA: 2.9
WE: Information Technology (Consulting)
A sphere is inscribed in a cube with an edge of 10. What is  [#permalink]

### Show Tags

7
1
86 00:00

Difficulty:   65% (hard)

Question Stats: 57% (01:38) correct 43% (01:47) wrong based on 660 sessions

### HideShow timer Statistics

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

_________________
Best Regards,
E.

MGMAT 1 --> 530
MGMAT 2--> 640
MGMAT 3 ---> 610
GMAT ==> 730

Originally posted by enigma123 on 12 Feb 2012, 21:48.
Last edited by Bunuel on 12 Feb 2012, 22:21, edited 1 time in total.
Edited the question
Math Expert V
Joined: 02 Sep 2009
Posts: 59720

### Show Tags

48
40
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: $$x=\frac{Diagonal -Diameter}{2}=\frac{10*\sqrt{3}-10}{2}=5(\sqrt{3}-1)$$

_________________
##### General Discussion
Manager  Status: Employed
Joined: 17 Nov 2011
Posts: 78
Location: Pakistan
GMAT 1: 720 Q49 V40 GPA: 3.2
WE: Business Development (Internet and New Media)
Re: A sphere is inscribed in a cube with an edge of 10. What is  [#permalink]

### Show Tags

2
Wicked Question and great simple explanation Bunuel!
Intern  Joined: 19 Mar 2011
Posts: 1
A sphere is inscribed in a cube with an edge of 10.  [#permalink]

### Show Tags

2
1
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)
Senior Manager  Joined: 06 Aug 2011
Posts: 317
Re: A sphere is inscribed in a cube with an edge of 10. What is  [#permalink]

### Show Tags

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?
_________________
Bole So Nehal.. Sat Siri Akal.. Waheguru ji help me to get 700+ score !
Math Expert V
Joined: 02 Sep 2009
Posts: 59720
Re: A sphere is inscribed in a cube with an edge of 10. What is  [#permalink]

### Show Tags

14
2
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 98514 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?
_________________
Intern  Joined: 09 May 2013
Posts: 43
Re: A sphere is inscribed in a cube with an edge of 10. What is  [#permalink]

### Show Tags

i dint understood why diameter is 10?
how smallest area is 1/2(diameter-diagonal) bunuel help
Math Expert V
Joined: 02 Sep 2009
Posts: 59720
Re: A sphere is inscribed in a cube with an edge of 10. What is  [#permalink]

### Show Tags

6
3
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 78968 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.
_________________
Intern  Joined: 09 May 2013
Posts: 43
Re: A sphere is inscribed in a cube with an edge of 10. What is  [#permalink]

### Show Tags

hi bunuel
thanks for an explanation.
why diagonal-diameter divided by 2
Math Expert V
Joined: 02 Sep 2009
Posts: 59720
Re: A sphere is inscribed in a cube with an edge of 10. What is  [#permalink]

### Show Tags

1
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...
_________________
Intern  Joined: 09 May 2013
Posts: 43
Re: A sphere is inscribed in a cube with an edge of 10. What is  [#permalink]

### Show Tags

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.

Posted from my mobile device
Intern  Joined: 17 Dec 2013
Posts: 5
Re: A sphere is inscribed in a cube with an edge of 10. What is  [#permalink]

### Show Tags

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)
Senior Manager  Joined: 08 Apr 2012
Posts: 323
Re: A sphere is inscribed in a cube with an edge of 10. What is  [#permalink]

### Show Tags

1
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?
Manager  Joined: 24 Oct 2012
Posts: 61
WE: Information Technology (Computer Software)
Re: A sphere is inscribed in a cube with an edge of 10. What is  [#permalink]

### Show Tags

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.
Manager  Joined: 07 Apr 2014
Posts: 98
Re: A sphere is inscribed in a cube with an edge of 10. What is  [#permalink]

### Show Tags

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.
Intern  Joined: 21 Apr 2014
Posts: 39
Re: A sphere is inscribed in a cube with an edge of 10. What is  [#permalink]

### Show Tags

1
1
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.
_________________
Eliza
GMAT Tutor
bestgmatprepcourse.com
Intern  Joined: 12 Jan 2015
Posts: 14
Concentration: Entrepreneurship, Human Resources
GMAT Date: 06-27-2015
Re: A sphere is inscribed in a cube with an edge of 10. What is  [#permalink]

### Show Tags

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?
EMPOWERgmat Instructor V
Status: GMAT Assassin/Co-Founder
Affiliations: EMPOWERgmat
Joined: 19 Dec 2014
Posts: 15716
Location: United States (CA)
GMAT 1: 800 Q51 V49 GRE 1: Q170 V170 Re: A sphere is inscribed in a cube with an edge of 10. What is  [#permalink]

### Show Tags

1
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)

GMAT assassins aren't born, they're made,
Rich
_________________
Board of Directors P
Joined: 17 Jul 2014
Posts: 2491
Location: United States (IL)
Concentration: Finance, Economics
GMAT 1: 650 Q49 V30 GPA: 3.92
WE: General Management (Transportation)
Re: A sphere is inscribed in a cube with an edge of 10. What is  [#permalink]

### Show Tags

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

oh man..almost got me tricked this one...

to find the shortest distance, we need to find the diagonal of the cube. we have a side of the cube = 10.
we can apply the Pythagorean theorem: x^2 = a^2 + b^2 + c^2 where a,b,c=10. or, we can just simply apply the 45-45-90 triangle property. anyways, the diagonal of the cube is 10*sqrt(3).
now, we know that the radius must be 5, and thus the diagonal must be 10.
from 10*sqrt(3), subtract 10 (diagonal).
this is not the end. the result will be the distance from both sides of the edges to the sphere.
we need to divide this by 2, and we get:
5*sqrt(3) - 5.
we can factor a 5, and get to D.
Manager  Joined: 06 Jun 2014
Posts: 85
Location: United States
Concentration: Finance, General Management
GMAT 1: 450 Q27 V21 GPA: 3.47
Re: A sphere is inscribed in a cube with an edge of 10. What is  [#permalink]

### Show Tags

2
Here is the official explanation.
Attachments cubeSphere.jpg [ 94.97 KiB | Viewed 53109 times ]

_________________
1) Kaplanprep 450 Q27 V21
2) Manhattan 530 Q35 V28
3) GmatPrep 450 Q33, V19
4) Veritas 460 Q31, V23
5) Veritas 440 Q 30, V21
6) Veritas 500 Q34, V 25
7) Gmat 420 Q27, V23
8) Veritas 520 Q36, V26 2/2
9) Veritas 540 Q37, V28 4/19
10)Manhattan 560 Q40, V28 4/28 Re: A sphere is inscribed in a cube with an edge of 10. What is   [#permalink] 30 Apr 2016, 19:43

Go to page    1   2    Next  [ 23 posts ]

Display posts from previous: Sort by

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