Question

How do you find the volume of the solid bounded by the coordinate planes and the plane $8x+6y+z=6$?

Answer 1

From the question given we have to find the volume of the solid bounded by the coordinate planes and the plane $8x+6y+z=6$. To solve the above question, we know that the given plane is tetrahedron and we also know that the volume of the tetrahedron is $v=\dfrac{1}{6}\left| \overrightarrow{a}\left( \overrightarrow{b}\times \overrightarrow{c} \right) \right|$. First, we have to find the vector $\overrightarrow{a}$, $\overrightarrow{b}$ and $\overrightarrow{c}$. For finding the vectors first we have to find the vertices and then by using the formula, we can find the volume of the above tetrahedron.

Complete step by step answer:
From the question given we have to find the volume of the solid bounded by the coordinate planes and the plane
$\Rightarrow 8x+6y+z=6$
we have a tetrahedron,
$\Rightarrow 8x+6y+z=6$
Now let us find the vertices of a tetrahedron,
First of all, let $y=0\ and\ z=0$
By substituting the values in the equation, we will get,
$\Rightarrow 8x=6$
By simplifying further, we will get,
$\Rightarrow x=\dfrac{3}{4}$
Therefore, the vertex
$\Rightarrow \overrightarrow{a}=\left[ \dfrac{3}{4},0,0 \right]$
now let $x=0\ and\ z=0$
By substituting the values in the equation, we will get,
$\Rightarrow 6y=6$
By simplifying further, we will get,
$\Rightarrow y=1$
Therefore, the vertex
$\Rightarrow \overrightarrow{b}=\left[ 0,1,0 \right]$
Now let $x=0\ and\ y=0$
By substituting the values in the equation, we will get,
$\Rightarrow z=6$
Therefore, the vertex
$\Rightarrow \overrightarrow{c}=\left[ 0,0,6 \right]$
As we know that the formula for tetrahedron is
$\Rightarrow v=\dfrac{1}{6}\left| \overrightarrow{a}\left( \overrightarrow{b}\times \overrightarrow{c} \right) \right|$
By substituting in the formula, we will get,
$\Rightarrow v=\dfrac{1}{6}\left| \begin{matrix} \dfrac{3}{4} & 0 & 0 \\\ 0 & 1 & 0 \\\ 0 & 0 & 6 \\\ \end{matrix} \right|$
$\Rightarrow v=\dfrac{1}{6}\times \dfrac{3}{4}\times 1\times 6$
By simplifying further, we will get,
$\Rightarrow v=\dfrac{3}{4}$

Therefore, the volume of the plane with the coordinate plane is $\dfrac{3}{4}$

Note: Students should know the basic formulas of tetrahedron, pyramid etc. the formula for the volume of regular tetrahedron with edge ‘a’ is $v=\dfrac{{{a}^{3}}}{6\sqrt{2}}$. The formula for volume of the pyramid is $\dfrac{1}{3}\times \left( \text{base area} \right)\times height$.