let a and b be opposite vertices of a unit cube (i.e. , the distance between a and b is the square root of 3). find the radius of a sphere, whose center is in the interior of the cube, that is tangent to the three faces that at a and also tangent to the three edges that meet at b.
To find the radius of the sphere, we need to consider the geometry of the problem. Let's break it down step by step.
1. Start by visualizing a unit cube in three dimensions. Place point A at one corner of the cube and point B at the opposite corner. The distance between A and B is the square root of 3.
2. Now imagine a sphere with its center inside the cube. We want the sphere to be tangent to three faces at point A and also tangent to three edges meeting at point B.
3. Since the sphere is tangent to the faces at point A, the radius of the sphere is perpendicular to these faces. This radius is also perpendicular to the line segment AB.
4. Note that the line segment AB is the space diagonal of the cube. In a cube, the space diagonal is the square root of 3 times the length of one side since the sides of the unit cube are all equal to 1.
5. Therefore, the radius of the sphere is perpendicular to AB and has a length equal to one-third of AB. Since AB is the square root of 3, the radius of the sphere is (1/3) * square root of 3.
Hence, the radius of the sphere, whose center is in the interior of the cube, that is tangent to the three faces at point A and also tangent to the three edges meeting at point B, is (1/3) * square root of 3.