Question

Find the volume of the tetrahedron with the given vertices. (4, 0, 0), (0, 4, 0), (0, 0, 1), (4, 4, 1)

Answer 1

`\mathbf{V = (16)/(3)}`

Explanation:

From the information given:

A = (4,0,0) B = (0,4,0) C = (0,0,1) D = (4,4,1)

`\overline{ AB}` = (0,4,0) - (4,0,0) = (-4, 4, 0)

`\overline{ BC} =` (0,0,1) - (0,4,0) = (0, -4, 1)

`\overline{ CD}` = (4,4,1) - (0,0,1) = (4, 4, 0)

The Volume of the tetrahedron
`V = (1)/(6) \bigg| AB * BC * CD \bigg|`

The Volume of the tetrahedron
`V = (1)/(6) \left\begin{vmatrix}-4,&4&0\\0,&-4&1\\4&4&0\end{vmatrix}\right`

`V = (1)/(6)\bigg [ -4(-4 *0 - 1*4) -4(0*0-1*4) +0(0*4--4*4) \bigg ]`

`V = (1)/(6)\bigg [ -4(0 - 4) -4(0-4) +0(0+16) \bigg ]`

`V = (1)/(6)\bigg [ 16 + 16 +0 \bigg ]`

`V = (1)/(6)\bigg [ 32 \bigg ]`

`\mathbf{V = (16)/(3)}`