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A boy would like to cut a cube out of a wooden sphere for his design and technology...

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phanpal999 | Student, Grade 8 | (Level 1) Salutatorian

Posted June 15, 2013 at 2:21 AM via web

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A boy would like to cut a cube out of a wooden sphere for his design and technology class.Given that the radius of the sphere is 8cm,find the possible dimension of the cube so that there will be a minimum wastage of wood?

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justaguide | College Teacher | (Level 2) Distinguished Educator

Posted June 15, 2013 at 3:53 AM (Answer #1)

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A cube has to be cut from a sphere of radius 8 cm such that the least amount of material is wasted. The largest cube that can be constructed such that it fits in the sphere is one that has all the 8 corners touching the sphere.

If this is the case, the distance of the center of the cube to any corner is equal to the radius of the sphere. Let the length of the side of the cube be x cm. The distance from the center of the cube to any corner is equal to `sqrt((sqrt((x/2)^2+(x/2)^2))^2 + (x/2)^2)`

= `sqrt(sqrt(2*(x/2)^2)^2 + (x/2)^2)`

= `sqrt(2*(x/2)^2 + (x/2)^2)`

= `sqrt(3*(x/2)^2)`

= `sqrt3*(x/2)`

As this is equal to the radius of the sphere:

`sqrt3*(x/2) = 8`

=> `x = 16/sqrt 3`

The side of the largest cube that can be created from the sphere is `16/sqrt 3`

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