The surface areas of the faces of a cuboid sharing a vertex are given as `25" m"^2, 32" m"^2 and 32" m"^2`. What is the volume of the cuboid?

Options

A. `160" m"^3`
B. `92" m"^3`
C. `184" m"^3`
D. `sqrt(3024)" m"^3`

Answer

To find the volume of the cuboid given the surface areas of the faces sharing a vertex, we can follow these steps:

### Step 1: Identify the relationships between the dimensions and surface areas
Let the dimensions of the cuboid be \( L \) (length), \( B \) (breadth), and \( H \) (height). The surface areas of the three faces sharing a vertex are given as:
- \( L \times B = 25 \, m^2 \) (1)
- \( B \times H = 32 \, m^2 \) (2)
- \( H \times L = 32 \, m^2 \) (3)

### Step 2: Multiply the three equations
We multiply equations (1), (2), and (3):
\[ (L \times B) \times (B \times H) \times (H \times L) = 25 \times 32 \times 32 \]
This simplifies to:
\[ L^2 \times B^2 \times H^2 = 25 \times 32 \times 32 \]

### Step 3: Calculate the right-hand side
Now, calculate \( 25 \times 32 \times 32 \):
\[ 25 \times 32 = 800 \]
\[ 800 \times 32 = 25600 \]
So, we have:
\[ L^2 \times B^2 \times H^2 = 25600 \]

### Step 4: Take the square root
Taking the square root of both sides gives:
\[ L \times B \times H = \sqrt{25600} \]

### Step 5: Simplify the square root
Now we simplify \( \sqrt{25600} \):
\[ \sqrt{25600} = \sqrt{256 \times 100} = \sqrt{256} \times \sqrt{100} = 16 \times 10 = 160 \]

### Step 6: Conclusion
Thus, the volume \( V \) of the cuboid is:
\[ V = L \times B \times H = 160 \, m^3 \]

### Final Answer
The volume of the cuboid is \( 160 \, m^3 \).

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