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.