The areas of three adjacent faces of a cuboid are a, b and c sq m. Its volume will be

Options

A. abc `m^(3)`
B. `sqrt(a^(2) + b^(2) + c^(2)) m^(3)`
C. `sqrt(abc) m^(3)`
D. `3 sqrt(abc) m^(3)`

Answer

To find the volume of a cuboid when the areas of three adjacent faces are given as \( a \), \( b \), and \( c \) square meters, we can follow these steps:

### Step 1: Understand the dimensions of the cuboid
Let the dimensions of the cuboid be:
- Length = \( L \)
- Breadth = \( B \)
- Height = \( H \)

The areas of the three adjacent faces can be expressed as:
- Area of face 1 (Length × Breadth) = \( A \) → \( L \times B = a \)
- Area of face 2 (Breadth × Height) = \( B \) → \( B \times H = b \)
- Area of face 3 (Height × Length) = \( C \) → \( H \times L = c \)

### Step 2: Set up the equations
From the above definitions, we can write the following equations:
1. \( L \times B = a \) (Equation 1)
2. \( B \times H = b \) (Equation 2)
3. \( H \times L = c \) (Equation 3)

### Step 3: Multiply the equations
Now, we will multiply all three equations together:
\[ (L \times B) \times (B \times H) \times (H \times L) = a \times b \times c \]
This simplifies to:
\[ L^2 \times B^2 \times H^2 = a \times b \times c \]

### Step 4: Take the square root
To find the volume \( V \) of the cuboid, we need to find \( L \times B \times H \):
\[ (L \times B \times H)^2 = a \times b \times c \]
Taking the square root of both sides gives:
\[ L \times B \times H = \sqrt{a \times b \times c} \]

### Step 5: Conclusion
The volume \( V \) of the cuboid is given by:
\[ V = L \times B \times H = \sqrt{a \times b \times c} \]

### Final Answer
Thus, the volume of the cuboid is \( \sqrt{a \times b \times c} \) cubic meters.

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