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

Question

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

Bunuel · Accepted Answer

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)

Consider an image of a sphere inscribed in a cube:
 
 https://gmatclub.com/gmat-practice-tests/resources/import/img/M25-08.png 
 
Let's assume that  x=10  centimeters.
 
Given that a sphere is inscribed in the cube, the edge length of the cube is equal to the sphere's diameter. Hence,  Diameter=10 .
 
The cube's diagonal is then calculated as  Diagonal=\sqrt{10^2+10^2+10^2}=10\sqrt{3} .
 
The distance between the vertex of the cube and the surface of the sphere is equal to half of the difference between the diagonal and the diameter. Therefore,  gap=\frac{Diagonal-Diameter}{2}=\frac{10*\sqrt{3}-10}{2}=5(\sqrt{3}-1) .
 
Given that  x=10 , we can express the gap as  5(\sqrt{3}-1)=\frac{x}{2}(\sqrt{3} - 1) .

Answer: D

KarishmaB · Answer

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.

skamal7 · Answer

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?

gmatzac · Answer

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

gmatzac · Answer

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.

pacifist85 · Answer

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...

reto · Answer

Does GMAC require this knowledge about spheres and these distances?

KarishmaB · Answer

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.

KarishmaB · Answer

Responding to a pm: 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). cube111.PNG D = \sqrt{d^2 + a^2} = \sqrt{{sqrt(10^2 + 10^2)}^2 + 10^2}} = \sqrt{10^2 + 10^2 + 10^2}

Bunuel · Answer

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 what-is-the-measure-of-the-angle-made-by-the-diagonals-of-th-88522.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

rohit8865 · Answer

download.png look at the fig. diagonal of cube= 3*x but sphere of radius x is in between cube so left out distance from two opposite vertices = 3*x-x but we need only any one side distance --->1/2( 3*x-x) x/2( 3*-1) Ans D

Fdambro294 · Answer

If we looked at a normal circle inscribed inside a 2-D Square, the Diameter of the Inscribed Circle would equal the Length of the Side of the square

In this case, the Diameter of the Sphere that would be perfectly inscribed inside a Cube of edge Length X would be ———> X = Diameter of the Sphere

If we drew the Main Diagonal of the Cube through the Center of the Cube from one vertex to its diagonally opposite vertex, the Length of this Main Diagonal would be ———> X * sqrt(3)

Along this Main Diagonal that traveled through the center of the Cube, it would pass through a “space” - then the Diameter of the Sphere - and then another “space”

These 2 “spaces” would be the distances from each diagonal opposite vertex to the surface or the Inscribed Sphere (along the main diagonal).

Since we are only looking for one distance, we can subtract the Diameter of the Sphere from the main diagonal and take (1/2) of the result:

(1/2) (X * sqrt(3) - X)

(X/2) (sqrt(3) - 1)

D

Posted from my mobile device

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

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

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