The areas of three consecutive faces of a cuboid are `12 cm^(2), 20 cm^(2)` and `15 cm^(2)`, then the volume (in `cm^(3)`) of the cuboid is

Options

A. 3600
B. 100
C. 80
D. 60

Answer

To find the volume of the cuboid given the areas of three consecutive faces, we can use the following steps:

### Step 1: Identify the areas of the faces
Let the areas of the three consecutive faces of the cuboid be:
- Area 1 (A1) = 12 cm²
- Area 2 (A2) = 20 cm²
- Area 3 (A3) = 15 cm²

### Step 2: Relate the areas to the dimensions
Let the dimensions of the cuboid be \( l \), \( b \), and \( h \) (length, breadth, and height). The areas of the faces can be expressed as:
- \( A1 = l \times b \)
- \( A2 = b \times h \)
- \( A3 = h \times l \)

### Step 3: Write equations based on the areas
From the areas given, we can write the following equations:
1. \( l \times b = 12 \) (1)
2. \( b \times h = 20 \) (2)
3. \( h \times l = 15 \) (3)

### Step 4: Multiply all three equations
To find the volume \( V \) of the cuboid, we can use the relationship:
\[ V = l \times b \times h \]
We can find \( V \) by multiplying the three equations:
\[ (l \times b) \times (b \times h) \times (h \times l) = A1 \times A2 \times A3 \]

This gives:
\[ (l^2 \times b^2 \times h^2) = 12 \times 20 \times 15 \]

### Step 5: Calculate the product of the areas
Now, calculate \( 12 \times 20 \times 15 \):
- \( 12 \times 20 = 240 \)
- \( 240 \times 15 = 3600 \)

So, we have:
\[ l^2 \times b^2 \times h^2 = 3600 \]

### Step 6: Take the square root to find the volume
Now, to find the volume \( V \):
\[ V^2 = 3600 \]
Taking the square root of both sides:
\[ V = \sqrt{3600} \]
\[ V = 60 \, \text{cm}^3 \]

### Final Answer
Thus, the volume of the cuboid is \( 60 \, \text{cm}^3 \).

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