The area of three adjacent faces of a cuboid are x, y, z square units respectively. If the volume of the cuboid be y cubic units, then the correct relation betewen v, x, y, z is

Options

A. a)`v^(2) = xyz`
B. b)`v^(3) = xyz`
C. c)`v^(2) = x^(3)y^(3)z^(3)`
D. d)`v^(3) = x^(2)y^(2)z^(2)`

Answer

To solve the problem, we need to establish a relationship between the volume \( V \) of the cuboid and the areas of its three adjacent faces, which are given as \( x \), \( y \), and \( z \).

### Step-by-Step Solution:

1. **Understand the Dimensions of the Cuboid**:
Let the dimensions of the cuboid be:
- Length = \( l \)
- Breadth = \( b \)
- Height = \( h \)

2. **Identify the Areas of the Adjacent Faces**:
The areas of the three adjacent faces of the cuboid can be expressed as:
- Area of face 1 (length and height) = \( l \times h = z \)
- Area of face 2 (length and breadth) = \( l \times b = x \)
- Area of face 3 (breadth and height) = \( b \times h = y \)

3. **Express the Volume of the Cuboid**:
The volume \( V \) of the cuboid is given by the formula:
\[ V = l \times b \times h \]

4. **Relate the Areas to the Dimensions**:
From the areas, we can express \( l \), \( b \), and \( h \) in terms of \( x \), \( y \), and \( z \):
- From \( l \times b = x \), we can express \( l \) as \( l = \frac{x}{b} \).
- From \( b \times h = y \), we can express \( h \) as \( h = \frac{y}{b} \).
- From \( l \times h = z \), we can express \( h \) as \( h = \frac{z}{l} \).

5. **Substitute to Find Volume**:
Substitute the expressions for \( l \) and \( h \) into the volume formula:
\[ V = l \times b \times h = l \times b \times \frac{y}{b} = l \times y \]
Now substitute \( l \) from \( l = \frac{x}{b} \):
\[ V = \frac{x}{b} \times y \]
Rearranging gives:
\[ V \times b = x \times y \]

6. **Establish the Relationship**:
Now, we can find the relationship between \( V \), \( x \), \( y \), and \( z \):
\[ V^2 = \sqrt{xyz} \]
Squaring both sides gives:
\[ V^2 = \frac{xyz}{b^2} \]
Thus, the relationship can be simplified to:
\[ V^2 = xyz \]

### Final Relation:
The correct relation between \( V \), \( x \), \( y \), and \( z \) is:
\[ V^2 = xyz \]