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

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01 Nov 2006, 08:50

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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?

My answer is d.
The length of the diagonal of the face of the cube will be
10sqrt(3).
The radius for sphere will be 5
Smallest distance will be =(10sqrt(3)/2)-5 =5(sqrt(3)-1)
What is OA?

i see a right triangle with the radii as two sides and the third side =5sqrt2, therefore the answer is 5sqrt2-5

what's wrong in my thinking?

The 2 sides, linked to the right angle, are :
> 5 (radius of the sphere)
> 10/2*sqrt(2) (the half part of line's length from 2 opposite vertex in a same side)

Thus, for the third, we have: x^2 = 25*2 + 25 = 25*3 that implies x = 5*sqrt(3)

I see your point Fig, but i still don't see why the other side should be 10/2*sqrt(2), where does the sqrt(2) come from?
If the sphere is inscribed then, it is touching the sides of the cube in the middle so the 2 sides of the right angle should be identical.

I see your point Fig, but i still don't see why the other side should be 10/2*sqrt(2), where does the sqrt(2) come from? If the sphere is inscribed then, it is touching the sides of the cube in the middle so the 2 sides of the right angle should be identical.

Appreciate your further explanation, thanks!!!

As u said, the contact point of the sphere is the center (gravity point) of a scare side. But, the distance is not the radius.

Actually, we can use the fact that the lenght of a line joining 2 opposite vertex on a same scare side is defined by hyp^2=10^2 + 10^2 => hyp = 10*sqrt(2).

Thus, to have the distance from 1 vertex to the contact point of the sphere at the center of a side, we divid by 2: hyp/2 = 10*sqrt(2)/2 = 5*sqrt(2)

The most simple is to draw 1 side of the cube... then, u can notice that each side measure 10.

I don't understand that last answer, so maybe it's right.

Anyway, here's my solution.

Imagine the cube sitting with one corner at the origin and another corner at (10, 10, 10). The distance from these two points is the distance of the long diagonal. The distance formula says that this distance is Sqrt((10 - 0)^2 + (10- 0)^2 + (10-0)^2) = Sqrt(3*10^2) = 10*Sqrt(3).

The sphere sitting inside the cube has diameter 10. Since a sphere in a cube is completely symmetric, the distance from a vertex to the sphere is the same for all vertices. So along that long diagonal we run through the middle of the sphere. The distance outside the sphere is 10*Sqrt(3) - 10. The distance from one vertex to the sphere must be half that by symmetry. So the answer is 1/2*(10*Sqrt(3) - 10) = 5*(Sqrt(3) - 1)

I haven't decided yet, i like Kellogg, LBS and INSEAD. My exam is in a few weeks and i want to see if i get a good score, otherwise everything changes!