The volume of a rectangular box whose area of three adjacent faces are 50 `cm ^2`, 30 `cm ^2` and 20 `cm ^2` is

Options

A. 600 `cm^3`
B. 1500 `cm^3`
C. 173 `cm^3`
D. 371 `cm^3`

Answer

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

### Step 1: Define the variables
Let:
- \( L \) = Length of the box
- \( B \) = Breadth of the box
- \( H \) = Height of the box

### Step 2: Write the equations based on the areas of the faces
From the problem, we know the areas of the three adjacent faces:
1. Area of face with dimensions \( L \) and \( H \):
\[ L \times H = 50 \quad \text{(1)} \]
2. Area of face with dimensions \( B \) and \( H \):
\[ B \times H = 30 \quad \text{(2)} \]
3. Area of face with dimensions \( L \) and \( B \):
\[ L \times B = 20 \quad \text{(3)} \]

### Step 3: Multiply all three equations
To find \( L \times B \times H \), we multiply all three equations:
\[ (L \times H) \times (B \times H) \times (L \times B) = 50 \times 30 \times 20 \]
This simplifies to:
\[ (LBH)^2 = 50 \times 30 \times 20 \]

### Step 4: Calculate the right-hand side
Now, calculate \( 50 \times 30 \times 20 \):
\[ 50 \times 30 = 1500 \]
\[ 1500 \times 20 = 30000 \]
Thus, we have:
\[ (LBH)^2 = 30000 \]

### Step 5: Take the square root
To find \( LBH \), we take the square root of both sides:
\[ LBH = \sqrt{30000} \]

### Step 6: Simplify the square root
We can simplify \( \sqrt{30000} \):
\[ \sqrt{30000} = \sqrt{300 \times 100} = \sqrt{300} \times 10 \]
Now, \( \sqrt{300} = \sqrt{100 \times 3} = 10\sqrt{3} \), so:
\[ LBH = 10 \times 10\sqrt{3} = 100\sqrt{3} \]

### Step 7: Calculate the approximate value
Using \( \sqrt{3} \approx 1.732 \):
\[ LBH \approx 100 \times 1.732 \approx 173.2 \]

### Conclusion
The volume of the rectangular box is approximately \( 173 \, \text{cm}^3 \).