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Which is the longest stick that can be contained inside a cube with side length 1 m?...

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Jackly | eNotes Newbie

Posted November 5, 2013 at 9:05 PM via web

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Which is the longest stick that can be contained inside a cube with side length 1 m?

Please show calculation!

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tiburtius | High School Teacher | (Level 3) Associate Educator

Posted November 5, 2013 at 9:27 PM (Answer #1)

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What you are searching is the longest line inside the cube which is of course space diagonal.

Formula for the length of space diagonal of a cuboid is

`d=sqrt(a^2+b^2+c^2)`

where `a,` `b` and `c` are lengths of the edges. Since cube has all equal sides formula for space diagonal of a cube is

`d=sqrt(a^2+a^2+a^2)=sqrt(3a^2)=asqrt3`

Hence if `a=1"m"` then 

`d=1sqrt3=sqrt3approx1.732`  ` `

The length of the longest stick that can be containdes in cube with side length of 1m is `sqrt3"m."`

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chrisyhsun | TA , College Freshman | (Level 1) Salutatorian

Posted August 13, 2014 at 5:04 PM (Answer #2)

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The longest stick that can be contained inside the cube will span the longest possible distance within the cube. This is the case when one end is at a given corner (let's call it A) and the other end is at the opposite corner. It may be a bit difficult to visualize what the "opposite corner" is on a cube, but if you position the cube so that vertex A is facing you head-on, the opposite will be the only corner you can't see from your vantage point.

Now, if you can "see" which two corners the stick spans, you'll be able to tell that this isn't just a problem of simple Pythagorean theorem. You actually have to essentially do the Pythagorean theorem twice - once "along" one face of the cube and another "along" a different face to get to the far corner. Since all of the edges are of length one, the calculation goes like this:

Pythagorean Theorem First Time:

1^2 + 1^2 = x^2

1 + 1 = x^2

2 = x^2 for now, we will keep it in this format, just to make the next calculation easier

Pythagorean Theorem Second Time: this time, though, instead of using two of the edges, we will use the answer from the previous calculation and another edge.

x^2 + 1^2 = y^2 (this means y is actually our final answer, not x!)

2 + 1 = y^2

3 = y^2

Taking the square root of both sides, we get that the length of the stick is square root of 3 meters.

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