The volume of a cube is `512 cm^(3)` . Find the length of its diagonal

Options

A. 8 cm
B. ` 8 sqrt(2)` cm
C. ` 8 sqrt(3)` cm
D. 16 cm

Answer

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

### Step 1: Understand the volume of a cube
The volume \( V \) of a cube is given by the formula:
\[ V = a^3 \]
where \( a \) is the length of one side of the cube.

### Step 2: Set up the equation
Given that the volume of the cube is \( 512 \, \text{cm}^3 \), we can set up the equation:
\[ a^3 = 512 \]

### Step 3: Solve for the side length \( a \)
To find \( a \), we need to take the cube root of both sides:
\[ a = \sqrt[3]{512} \]
Calculating the cube root:
\[ a = 8 \, \text{cm} \]

### Step 4: Find the diagonal of the cube
The formula for the diagonal \( d \) of a cube is given by:
\[ d = a \sqrt{3} \]
Substituting the value of \( a \):
\[ d = 8 \sqrt{3} \, \text{cm} \]

### Final Answer
The length of the diagonal of the cube is:
\[ d = 8 \sqrt{3} \, \text{cm} \]

---