The areas of three adjacent faces of a cuboid are `30 cm^(2), 20 cm^(2)` and `24 cm^(2)` . The volume of the cuboid is :

Options

A. `200 cm^(3)`
B. `120 cm^(3)`
C. `180 cm^(3)`
D. `150 cm^(3)`

Answer

To find the volume of the cuboid given the areas of three adjacent faces, we can follow these steps:

### Step 1: Define the dimensions of the cuboid
Let the dimensions of the cuboid be \( l \), \( b \), and \( h \). The areas of the three adjacent faces can be represented as:
- Area of face 1: \( l \times b = 30 \, \text{cm}^2 \)
- Area of face 2: \( b \times h = 20 \, \text{cm}^2 \)
- Area of face 3: \( h \times l = 24 \, \text{cm}^2 \)

### Step 2: Set up the equations
From the areas, we can set up the following equations:
1. \( lb = 30 \) (Equation 1)
2. \( bh = 20 \) (Equation 2)
3. \( hl = 24 \) (Equation 3)

### Step 3: Multiply all three equations
To find the volume \( V \) of the cuboid, we can use the formula:
\[ V = l \times b \times h \]
We can find \( (lbh)^2 \) by multiplying all three equations:
\[ (lb) \times (bh) \times (hl) = (lbh)^2 \]
Substituting the values from the equations:
\[ 30 \times 20 \times 24 = (lbh)^2 \]

### Step 4: Calculate the product
Now, calculate the product:
\[ 30 \times 20 = 600 \]
\[ 600 \times 24 = 14400 \]
So, we have:
\[ (lbh)^2 = 14400 \]

### Step 5: Take the square root
To find \( lbh \), we take the square root of both sides:
\[ lbh = \sqrt{14400} \]
Calculating the square root:
\[ lbh = 120 \, \text{cm}^3 \]

### Conclusion
The volume of the cuboid is \( 120 \, \text{cm}^3 \).