Question

If areas of three adjacent faces of a cuboid are \[x\], \[y\], and \[z\] respectively, then find the volume of the cuboid.

Answer 1

Hint:
Here, we need to find the volume of the cuboid. We will write the areas of the three adjacent faces in terms of the length, breadth, and height. Then, we will use the formula for the volume of a cuboid to find the required volume.

Formula used:
The volume of a cuboid is given by \[l \times b \times h\], where \[l\] is the length, \[b\] is the breadth, and \[h\] is the height.

Complete step by step solution:
Let \[l\] be the length of the cuboid, \[b\] be the breadth of the cuboid, and \[h\] be the height of the cuboid.
The areas of three adjacent faces of a cuboid are \[x\], \[y\], and \[z\] respectively.
The faces of a cuboid are in shape of a rectangle.
Now, we can find the areas of the faces of the cuboid.
Area of the face with length and breadth of cuboid as sides \[ = lb\]
Using the given information, we can rewrite this to get the equation
\[ \Rightarrow x = lb\]
Area of the face with breadth and height of cuboid as sides \[ = bh\]
Using the given information, we can rewrite this to get the equation
\[ \Rightarrow y = bh\]
Area of the face with length and height of cuboid as sides \[ = lh\]
Using the given information, we can rewrite this to get the equation
\[ \Rightarrow z = lh\]
We get the equations \[x = lb\], \[y = bh\], and \[z = lh\].
Therefore, multiplying the corresponding sides of the three equations, we get
\[ \Rightarrow xyz = lb \times bh \times lh\]
Rewriting the expression using exponents, we get
\[\begin{array}{l} \Rightarrow xyz = {l^2}{b^2}{h^2}\\ \Rightarrow xyz = {\left( {lbh} \right)^2}\end{array}\]
Taking the square root on both the sides, we get
\[ \Rightarrow \sqrt {xyz} = lbh\]
Now, we need to find out the volume of the given cuboid.
Thus, we get
Volume of the given cuboid \[ = lbh\]
Substituting \[lbh = \sqrt {xyz} \] in the formula, we get
Volume of the given cuboid \[ = \sqrt {xyz} \]

Therefore, we get the volume of the given cuboid as \[\sqrt {xyz} \].

Note:
We used the area of a rectangle in the solution. A rectangle is a quadrilateral whose opposite sides are equal and parallel, and the diagonals are equal in length. Each interior angle of a rectangle is exactly equal to 90 degrees. The area of a rectangle is the product of any two adjacent sides of the rectangle.