Math questions from any Manhattan Prep GMAT Computer Adaptive Test.
cavalli27
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Problem Solving: SPHERE IN A CUBE

by cavalli27 Thu Oct 09, 2014 8:04 am

Hi tutors,

Heres the question from one of my CAT's

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?

Heres the part that im having trouble understanding- perhaps i didnt know this rule or maybe cant connect with what iv studied before and how to apply it in this problem

Like the sides of the circle in the diagram above, the sides of a sphere inscribed in a cube will touch the sides of the cube. Therefore, a sphere inscribed in a cube will have a radius equal to half the length of the side of that cube.

I understand that to find the shortest distance (as you explained in your diagram in explanations), one can to subtract the radius of the sphere from 1/2 the length of the cubes diagonal . I also understand that 10root3 is the cubes diagonal and that 5root3 is the length of 1/2 the diagonal. I do NOT understand how to calculate the radius of the sphere? Can you explain to me the rule again and whether this rule applies to other objects too? I have all your study guides and have read rules on inscribed and circumscribed triangles/circles.

Thanks very much :)
RonPurewal
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Re: Problem Solving: SPHERE IN A CUBE

by RonPurewal Fri Oct 10, 2014 4:22 am

No "rules" needed here. Just picture a sphere crammed into a cubical box that fits it exactly. The sphere touches all the sides of the box; no wiggle room.

Since the sphere touches the box on the top and bottom, the diameter is the same as the height of the cube.
cavalli27
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Re: Problem Solving: SPHERE IN A CUBE

by cavalli27 Thu Nov 06, 2014 6:37 am

Thanks i saw your reply earlier but forgot to reply! :)
RonPurewal
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Re: Problem Solving: SPHERE IN A CUBE

by RonPurewal Thu Nov 06, 2014 4:25 pm

you're welcome.