M25-08

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)

Bunuel · Accepted Answer

Official Solution:

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

Bunuel · Answer

temoalta · Answer

I understand now. I thought that vertices were the edges of the cube. Thanks.

fsalek · Answer

I think this is a  high-quality  question and I agree with explanation.

Vinayprajapati · Answer

The sphere and cube arrangement will be as shown in figure.

1/2(Diagonal-Diameter) = 1/2 (a sqrt3 -a)= a/2 (sqrt3-1)

C3BISHT · Answer

I think this is a  high-quality  question and I agree with explanation.

coreyander · Answer

I think this is a  high-quality  question and I agree with explanation.

Bunuel · Answer

I have edited the question and the solution by adding more details to enhance its clarity. I hope it is now easier to understand.

M25-08

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