Question

How do you find the volume of the solid bounded by the coordinate planes and the plane $3x + 2y + z = 6$?

Answer 1

Here, we are given the solid bounded by the coordinate planes and the plane $3x + 2y + z = 6$ which is a tetrahedron. First, we will find the vertices of this tetrahedron. After that will use the formula of the volume of the tetrahedron by using its vertices to get our answer.

Complete step by step answer:
We are given the plane $3x + 2y + z = 6$.
Let us now find the vertices of the solid.
First, we will find the vertex in x direction.
To find the vertex in the x direction, we will put $y = 0$ and $z = 0$.
$3x + 2y + z = 6 \\\ \Rightarrow 3x = 6 \\\ \Rightarrow x = 2 \\\$
Therefore, vertex $\overrightarrow a = \left\langle {2,0,0} \right\rangle$.
Now, we will find the vertex in y direction.
To find the vertex in the y direction, we will put $x = 0$ and $z = 0$.
$3x + 2y + z = 6 \\\ \Rightarrow 2y = 6 \\\ \Rightarrow y = 3 \\\$
Therefore, vertex $\overrightarrow b = \left\langle {0,3,0} \right\rangle$.
Now, we will find the vertex in z direction.
To find the vertex in the z direction, we will put $x = 0$ and $y = 0$.
$3x + 2y + z = 6 \\\ \Rightarrow z = 6 \\\$
Therefore, vertex $\overrightarrow c = \left\langle {0,0,6} \right\rangle$.
We know that the formula for finding the volume using the vertices of the tetrahedron is given by:
$V = \dfrac{1}{6}\left| {\overrightarrow a \cdot \left( {\overrightarrow b \times \overrightarrow c } \right)} \right|$
We will now put the values of each of the vertices $\overrightarrow a = \left\langle {2,0,0} \right\rangle$, $\overrightarrow b = \left\langle {0,3,0} \right\rangle$ and $\overrightarrow c = \left\langle {0,0,6} \right\rangle$.

2&0&0 \\\ 0&3&0 \\\ 0&0&6 \end{array}} \right| = \dfrac{1}{6} \times 2 \times 3 \times 6 = 6uni{t^3}$$. **Thus our final answer is: The volume of the solid bounded by the coordinate planes and the plane $3x + 2y + z = 6$ is $$6uni{t^3}$$.** **Note:** Here, we have used the formula for the volume $$V = \dfrac{1}{6}\left| {\overrightarrow a \cdot \left( {\overrightarrow b \times \overrightarrow c } \right)} \right|$$. We can see that this involves the cross product which is a vector quantity. However, our final answer will be a scalar product as we have determined here because it also involves dot product in the formula.