The diagonal of a cube is `sqrt(192) cm`. Its volume (in `cm^(3)`) will be

Options

A. 216
B. 432
C. 512
D. 624

Answer

To find the volume of a cube given its diagonal, we can follow these steps:

### Step 1: Understand the relationship between the diagonal and the side length of the cube.
The formula for the diagonal \(d\) of a cube with side length \(a\) is given by:
\[ d = a\sqrt{3} \]

### Step 2: Set the diagonal equal to the given value.
We know from the problem that the diagonal \(d\) is \(\sqrt{192}\) cm. Therefore, we can set up the equation:
\[ a\sqrt{3} = \sqrt{192} \]

### Step 3: Solve for the side length \(a\).
To find \(a\), we can rearrange the equation:
\[ a = \frac{\sqrt{192}}{\sqrt{3}} = \sqrt{\frac{192}{3}} \]
Calculating \(\frac{192}{3}\):
\[ \frac{192}{3} = 64 \]
Thus, we have:
\[ a = \sqrt{64} = 8 \text{ cm} \]

### Step 4: Calculate the volume of the cube.
The volume \(V\) of a cube is given by:
\[ V = a^3 \]
Substituting the value of \(a\):
\[ V = 8^3 = 512 \text{ cm}^3 \]

### Conclusion:
The volume of the cube is \(512 \text{ cm}^3\).

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