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(sqrt3 â€“ 1)
C. 10(sqrt2 â€“ 1)
D. 5(sqrt3 â€“ 1)
E. 5(sqrt2 â€“ 1)
The needed distance lies on the line connecting two vertices which are symmetric through the centre of the sphere( this centre is also that of the cube)
Let D be the distance between the two symmetric vertices and d be the needed distance:
We have : 2*d= D - the diameter of the sphere
D can be calculated = sqrt( 10^2 + [10sqrt2]^2 ) = 10 sqrt3
the diameter of the sphere= 10 since the sphere is tangently inscribed in the cube
---> 2d= 10sqrt3 - 10 ---> d= 5( sqrt3-1)
D it is.