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Geometry question

by prernamalhotra » Fri May 30, 2014 6:35 am
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?

1)10(√3 - 1)

2) 5

3) 10(√2 - 1)

4) 5(√3 - 1)

5) 5(√2 - 1)

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by GMATGuruNY » Fri May 30, 2014 7:09 am
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)
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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.
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by [email protected] » Fri May 30, 2014 10:48 am
Hi Prerna,

Mitch's solution correctly explains the math behind this question, so I won't rehash it here. I will add a few details to what Mitch did with his approach though.

First, when dealing with a "rectangular solid" (a box or cube), the longest distance between any two points on the surface of the rectangular solid is actually found by going THROUGH the solid.

eg. Upper-Right-Back corner to Lower-Left-Front corner

You can calculate that distance with this formula:

Sq. Root(Length^2 + Width^2 + Height^2)

-------------
In this question, all 3 dimensions of the cube are 10, so the diagonal through the cube = Sq.Root(10^2 + 10^2 + 10^2) = Sq.Root(300) = 10√3.

Since the sphere was inscribed in the cube, the middle of the sphere is also the middle of the cube. Also, the diameter of the sphere = side length of the cube = 10. Thus, the sphere's radius = 5.

So we can subtract the radius of the sphere (5) from from half the diagonal line (5√3) to find the length from the corner to the sphere. The answer choices expect us to factor out a 5.

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