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November 21st, 2014, 06:16 AM   #1
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Question Spacediagonal 30 cm in a cube? Calculate the cubes side lengths?

Hi! I have this question about an exercise in my Swedish math book. It told me to calculate the cube's side lengths. But the only thing I know is the spacediagonal. Or whatever it's called in English.

I think that it has something to do with Pythagoras theorem. First I thought that it was that I could calculate the cube sides length using:

x^2 + x^2 = 30^2

But then I realized that the sides of the triangle isn't equal. So then I thought:

a^2 + b^2 = 30^2

And after that I got stuck.



We can say that d = 30 in this image. Do you get it? I really appreciate help!

Last edited by skipjack; November 21st, 2014 at 06:18 PM.
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November 21st, 2014, 08:22 AM   #2
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If a is the length of the edge of the cube, then $\displaystyle \ a\sqrt{3} \ $ is the length of the space diagonal of that cube.

Source:
Space diagonal - Wikipedia, the free encyclopedia


Set $\displaystyle \ a\sqrt{3} \ = \ 30, \ $ and solve for a.
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November 21st, 2014, 08:44 AM   #3
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Quote:
Originally Posted by DecoratorFawn82 View Post
Hi! I have this question about a exercise in my Swedish math book. It told me to calculate the cubes side lengths. But the only thing I know is the spacediagonal. Or whatever it's called in English.
In Swedish, the text of this problem

would perhaps be clearer.

So, you do that !
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November 21st, 2014, 09:36 AM   #4
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Quote:
Originally Posted by Math Message Board tutor View Post
If a is the length of the edge of the cube, then $\displaystyle \ a\sqrt{3} \ $ is the length of the space diagonal of that cube.

Source:
Space diagonal - Wikipedia, the free encyclopedia


Set $\displaystyle \ a\sqrt{3} \ = \ 30, \ $ and solve for a.
a sqrt(3) = 30
a = 30/(sqrt(3))
a = 17

Thanks! Your answer seem to be correct according to my facit in my math book . But could you please explain more in detail what exactly you are doing?

Where do you get sqrt(3) from? The volume of a cube x^3? Or that it is a triangle and it has 3 sides? And why would you take the square root of it?

I understand that a is the sides of the cube that we are going to calculate. But I do not really understand the sqrt(3).

Could you please explain?

EDIT:

I saw that you wrote that it is the length of the space diagonal. But could you please explain more in detail?

Last edited by skipjack; November 21st, 2014 at 06:22 PM.
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November 21st, 2014, 12:11 PM   #5
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Hello, DecoratorFawn82!

Quote:
The length of the space diagonal is 30.
Find the length of the side of the cube.


Look at the "floor" of the cube.

We have an $x\text{-by-}x$ square.

Code:
             x
      * - - - - - *
      | *          |
      |   *        |
      |     *      | x
      |       *    |
      |         *  |
      |           *|
      * - - - - - *
Its diagonal is $x\sqrt{2}.$


Look at the right triangle with the space diagonal.
Code:
      *
      |  *
      |     *   30
   x |        *
      |           *
      |              *
      * - - - - - - - - *
             x√2
We have: $\:x^2 + (x\sqrt{2})^2 \:=\:30^2$

Solve for $x.$

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November 21st, 2014, 01:42 PM   #6
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Quote:
Originally Posted by soroban View Post
Hello, DecoratorFawn82!


Look at the "floor" of the cube.

We have an $x\text{-by-}x$ square.

Code:
             x
      * - - - - - *
      | *          |
      |   *        |
      |     *      | x
      |       *    |
      |         *  |
      |           *|
      * - - - - - *
Its diagonal is $x\sqrt{2}.$


Look at the right triangle with the space diagonal.
Code:
      *
      |  *
      |     *   30
   x |        *
      |           *
      |              *
      * - - - - - - - - *
             x√2
We have: $\:x^2 + (x\sqrt{2})^2 \:=\:30^2$

Solve for $x.$

Is a diagonal of a cube always x*sqrt(2)?

So if x where 4 then it would be:

4*sqrt(2)?

Or where do you get sqrt(2) from?

Last edited by greg1313; November 21st, 2014 at 02:09 PM.
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November 21st, 2014, 06:27 PM   #7
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By Pythagoras, diagonal² = x² + x² = 2x², so diagonal = √2x.
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