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# HELP PLZ!!!!

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A cube has side 6 lengths. Its vertices are alternately colored black and purple, as shown below. What is the volume of the tetrahedron whose corners are the purple vertices of the cube? (A tetrahedron is a pyramid with a triangular base.)

Jun 21, 2018

#1
+8170
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Let  a  be the length of an edge of the tetrahedron.

A face of the cube looks like this:

By the Pythagorean theorem,

62 + 62  =  a2

Combine like terms....  n + n = n * 2    so    62 + 62  =  62 * 2

62 * 2  =  a2

Take the positive square root of both sides of the equation.

√[ 62 * 2 ]  =  a

√62  *  √2  =  a

6√2  =  a

Since each face of the cube is the same, each edge of the tetrahedron is the same.

So the tetrahedron is a regular tetrahedron.

The formula for the volume of a regular tetrahedron is

volume  =  (edge length)3 / ( 6√2 )

Plug in  6√2  for the edge length.

volume  =  ( 6√2 )3 / ( 6√2 )

Simplify.

volume  =  ( 6√2 )( 6√2 )( 6√2 ) / ( 6√2 )

volume  =  ( 6√2 )( 6√2 )

volume  =  36 * 2

volume  =  72      cubic units

Jun 21, 2018
#2
+22343
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A cube has side 6 lengths. Its vertices are alternately colored black and purple, as shown below.
What is the volume of the tetrahedron whose corners are the purple vertices of the cube?
(A tetrahedron is a pyramid with a triangular base.)

A cube has side 6 lengths. $$\mathbf{\text{Let s = 6}}$$

The cube has volume $$\mathbf{V_{\text{cube}} = s^3}$$

The 4 right triangular pyramids that must be carved off the cube to produce the regular tetrahedron each have volume

$$\mathbf{V_{\text{right triangular pyramid}} = \frac13\ \times \frac{s^2}{2} \times s }$$

The volume of the regular tetrahedron is  $$\mathbf{ V_{\text{cube} }- 4\times V_{\text{right triangular pyramid}} }$$

$$\begin{array}{|rcll|} \hline && V_{\text{tetrahedron}} \\\\ &=& V_{\text{cube} }- 4\times V_{\text{right triangular pyramid}} \\\\ &=& s^3 - 4\times \dfrac13\ \times \dfrac{s^2}{2} \times s \\\\ &=& s^3 - \dfrac23 s^3 \\\\ &=& \dfrac13 s^3 \quad & | \quad s = 6 \\\\ &=& \dfrac{6^3}{3} \\\\ &=& \dfrac{216}{3} \\\\ &=& 72 \\ \hline \end{array}$$

The volume of the tetrahedron is 72

Jun 21, 2018