Problem Solving

Unable to get the technique. Even POE not working.
Please help?

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(√3 - 1)
B.5
C.10(√2 - 1)
D.5(√3 - 1)
E.5(√2 - 1)

(my initial solution had a mistake so I rewrote it)

The shortest distance from a corner to the surface of the sphere will be the line that goes straight towards the center of the sphere. This line segment is aligned with the diagonal of the cube (distance between two opposite vertices - see figure 1 below).

Imagine taking a vertical cross-section of the figure. We'll end up with the 2nd figure below. The line segment labeled x is the distance we want. Notice that in these figures, the diagonal is not the diagonal of a square, but of a box. it follows the formula length^2 + width^2 + height^2 = diag^2




The answer is D.

-Patrick
GMATFix

I was just talking about a problem like this with one of my students last night!

The hardest part of this is visualizing how the shapes relate to each other. Since the sphere is inscribed, that means that the surface of the sphere touches each face of the cube at the center of the face. Thus, the sphere has a diameter of 10 since the cube has an edge of 10.

The shortest distance is diagonally from one of the vertices of the cube. To find the full length of the diagonal (from one vertex to the exact opposite vertex, i.e. front left lower corner to back right upper corner), we can use the formula D = sqrt(3s^2), which gives us D = sqrt(3*10^2) = sqrt(300) = 10sqrt3. All we need is half of this, though: the distance from the vertex to the center of the cube (which is also the center of the circle) is 5sqrt3.

In a sphere, the diameter is the same in any dimension, so, along the diagonal that we found, the diameter must be 10, giving us a radius of 5.

The distance from the center of the sphere to its surface is 5, and the distance from the center of the sphere to the vertex is 5sqrt3. Thus, the distance between the vertex and the surface is 5sqrt3 - 5, or 5(sqrt3 - 1). D

alekhya615 wrote:Unable to get the technique. Even POE not working.
Please help?

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(√3 - 1)
B.5
C.10(√2 - 1)
D.5(√3 - 1)
E.5(√2 - 1)

The formula for the diagonal of a cube = √(3e²).

In the figure above, x = the distance between the cube and the surface of the sphere.
The diagonal of the cube = 2x + the diameter of the sphere.
Thus, x = (diagonal of the cube - diameter of the sphere)/2.

The diagonal of the cube = √(3e²) = √(3*10²) = 10√3.
The diameter of the sphere = the edge of the cube = 10.
Thus, x = (10√3 - 10)/2 = 5√3 - 5 = 5(√3 - 1).

The correct answer is D.