Surface areas of three adjacent faces of a cuboid are p, q, r. Its volume is

Options

A. `sqrt(p q^(2) + q r^(2) + r p^(2))`
B. `(sqrt(pq) + sqrt(qr) + sqrt(rp))(p^(2) + q^(2) + r^(2))`
C. `(sqrt((p^(2) + q^(2) + r^(2))(p + q + r)))`
D. `sqrt(pqr)`

Answer

To find the volume of a cuboid when the surface areas of three adjacent faces are given as \( p \), \( q \), and \( r \), we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Surface Areas**:
- Let the dimensions of the cuboid be \( L \) (length), \( B \) (breadth), and \( H \) (height).
- The surface areas of the three adjacent faces are given as:
- \( L \times B = p \)
- \( B \times H = q \)
- \( H \times L = r \)

2. **Multiply the Surface Area Equations**:
- Multiply the three equations together:
\[ (L \times B) \times (B \times H) \times (H \times L) = p \times q \times r \]
- This simplifies to:
\[ L^2 \times B^2 \times H^2 = p \times q \times r \]

3. **Take the Square Root**:
- To find \( L \times B \times H \), take the square root of both sides:
\[ L \times B \times H = \sqrt{p \times q \times r} \]

4. **Volume of the Cuboid**:
- The volume \( V \) of the cuboid is given by the product of its dimensions:
\[ V = L \times B \times H \]
- Therefore, substituting the expression we found:
\[ V = \sqrt{p \times q \times r} \]

### Final Answer:
The volume of the cuboid is \( \sqrt{p \times q \times r} \).

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