Question

How do you find the volume of the solid that lies in the first octant and is bounded by the three quadrant planes and another plane passing through \[\left( 3,0,0 \right)\],\[\left( 0,4,0 \right)\] and \[\left( 0,0,5 \right)\]?

Answer 1

Hint: From the question we have been asked to find the volume of the solid formed by the three quadrant planes and another plane passing through given three points. So, for solving this question we will find what kind of solid it is and we will use the scalar triple product formula in vector algebra and find the volume of the solid.

Complete step by step solution:
Firstly, the solid in the given question is a tetrahedron, having the vertices of the tetrahedron as A,B,C.
Let the volume of the tetrahedron be as V.
So, the points are O\[\left( 0,0,0 \right)\], A\[\left( 3,0,0 \right)\], B\[\left( 0,4,0 \right)\], C\[\left( 0,0,5 \right)\].
The formula for finding the volume of a tetrahedron is as follows.
\[\Rightarrow v=\left| \dfrac{1}{6}\left[ \overrightarrow{OA,}\overrightarrow{OB,}\overrightarrow{OC} \right] \right|\]
Where, \[\left[ \overrightarrow{VA,}\overrightarrow{VB,}\overrightarrow{VC} \right]\] denotes the scalar triple product in the vector algebra, which can be written as follows.
\[\Rightarrow \overrightarrow{VA}\centerdot \left\{ \overrightarrow{VB}\times \overrightarrow{VC} \right\}\]
Now, we will find the \[\overrightarrow{OA}\] as follows.
We know,
\[\Rightarrow \overrightarrow{OA}=A(3,0,0)-O(0,0,0)=(3,0,0)\]
In the similar way we find the remaining two as follows.
We get,
\[\Rightarrow \overrightarrow{OB}=B(0,4,0)-O(0,0,0)=(0,4,0)\]
\[\Rightarrow \overrightarrow{OC}=C(0,0,5)-O(0,0,0)=(0,0,5)\]
So, now we substitute this in the formula of the volume of tetrahedron which is mentioned above, we get,
\[\Rightarrow v=\left| \dfrac{1}{6}\left[ \overrightarrow{OA,}\overrightarrow{OB,}\overrightarrow{OC} \right] \right|\]
\[\Rightarrow v=\left| \dfrac{1}{6}\left[ \left| \begin{align}
  & 3\text{ 0 0} \\
 & \text{0 4 0} \\
 & \text{0 0 5} \\
\end{align} \right| \right] \right|\]
\[\Rightarrow v=\dfrac{1}{6}\times 3\times 4\times 5\]
\[\Rightarrow v=10\]

Therefore, the volume of the solid given in the question is \[v=10\] cube units.

Note: Students must be very careful in doing the calculations. Students must have good knowledge in the concept of vector algebra. We should know the formula of the scalar triple product which is for \[\left[ \overrightarrow{VA,}\overrightarrow{VB,}\overrightarrow{VC} \right]\] the scalar triple product will be as follows.
\[\Rightarrow \overrightarrow{VA}\centerdot \left\{ \overrightarrow{VB}\times \overrightarrow{VC} \right\}\].