यदि घनाभ के तीन पृष्ठों का क्षेत्रफल क्रमश: x, y तथा z है, तब इस घनाभ का आयतन क्या होगा?

Options

A. `x+y+z`
B. `2(xy+yz+zx)`
C. `sqrt(xyz)`
D. `x^(2)y^(2)`

Answer

**Step-by-Step Solution:**

1. **Understanding the Problem:**
- We have a cuboid (घनाभ) with three adjacent faces having areas \(x\), \(y\), and \(z\). Let's denote the dimensions of the cuboid as \(a\), \(b\), and \(c\).
- The areas of the three faces can be expressed as:
- Area of face 1 (ab) = \(x\)
- Area of face 2 (bc) = \(y\)
- Area of face 3 (ca) = \(z\)

2. **Setting Up the Equations:**
- From the areas, we can write the following equations:
\[ ab = x \quad (1) \]
\[ bc = y \quad (2) \]
\[ ca = z \quad (3) \]

3. **Finding the Volume:**
- The volume \(V\) of the cuboid is given by the product of its dimensions:
\[ V = abc \]
- To find \(abc\), we can multiply equations (1), (2), and (3):
\[ (ab)(bc)(ca) = xyz \]
- This simplifies to:
\[ a^2b^2c^2 = xyz \]

4. **Taking the Square Root:**
- To find \(abc\), we take the square root of both sides:
\[ (abc)^2 = xyz \]
\[ abc = \sqrt{xyz} \]

5. **Final Answer:**
- Therefore, the volume \(V\) of the cuboid is:
\[ V = \sqrt{xyz} \]