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?
10( root(3)- 1)
5
10( root(2)- 1)
5( root(3)- 1)
5( root(2)- 1)
Explain pls!
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Sphere inscribed in a cube
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rijul007
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The sphere that would completely fit the cube with an edge of 10 will have a diameter of 10
and the figure can be represented like this
Acc to the figure, the shortest possible distance would be
[AB - (diameter of the sphere)]/2
[10*root(3) - 10]/2
5[root(3)-1]
Correct ans- 4th option
and the figure can be represented like this
Acc to the figure, the shortest possible distance would be
[AB - (diameter of the sphere)]/2
[10*root(3) - 10]/2
5[root(3)-1]
Correct ans- 4th option
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smackmartine
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IMO D
I would say this is more of a visualization problem. Because the diameter of the sphere is a part of the diagonal of the cube,in words , calculate the diagonal of the cube, subtract the diameter of the sphere and divide it by 2.
mathematically,
calculate the diagonal of the cube --> [(10)^2 + (10sqrt(2))^2]^1/2 = 10sqrt(3)
subtract the diameter of the sphere-> 10sqrt(3)-10
divide the result by 2 --> 5( root(3)- 1)
I would say this is more of a visualization problem. Because the diameter of the sphere is a part of the diagonal of the cube,in words , calculate the diagonal of the cube, subtract the diameter of the sphere and divide it by 2.
mathematically,
calculate the diagonal of the cube --> [(10)^2 + (10sqrt(2))^2]^1/2 = 10sqrt(3)
subtract the diameter of the sphere-> 10sqrt(3)-10
divide the result by 2 --> 5( root(3)- 1)
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GMATGuruNY
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I received a PM asking me to clarify the solution.GmatKiss 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?
10( root(3)- 1)
5
10( root(2)- 1)
5( root(3)- 1)
5( root(2)- 1)
Explain pls!
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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Followed here and elsewhere by over 1900 test-takers.
I have worked with students based in the US, Australia, Taiwan, China, Tajikistan, Kuwait, Saudi Arabia -- a long list of countries.
My students have been admitted to HBS, CBS, Tuck, Yale, Stern, Fuqua -- a long list of top programs.
As a tutor, I don't simply teach you how I would approach problems.
I unlock the best way for YOU to solve problems.
For more information, please email me (Mitch Hunt) at [email protected].
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