If the volume of a cube is `375 sqrt(3) cm^(3)` , then its diagonal is

Options

A. 10 cm
B. 15 cm
C. 14 cm
D. 12 cm

Answer

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

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

### Step 2: Set up the equation using the given volume.
We know the volume of the cube is \( 375 \sqrt{3} \, \text{cm}^3 \). Therefore, we can set up the equation:
\[ a^3 = 375 \sqrt{3} \]

### Step 3: Solve for the side length \( a \).
To find \( a \), we need to take the cube root of both sides:
\[ a = \sqrt[3]{375 \sqrt{3}} \]

### Step 4: Simplify the expression for \( a \).
To simplify \( \sqrt[3]{375 \sqrt{3}} \), we can break it down:
- First, factor \( 375 \):
\[ 375 = 125 \times 3 = 5^3 \times 3 \]
Thus, we can rewrite:
\[ \sqrt[3]{375 \sqrt{3}} = \sqrt[3]{5^3 \times 3 \times \sqrt{3}} = \sqrt[3]{5^3} \times \sqrt[3]{3 \sqrt{3}} = 5 \times \sqrt[3]{3^{3/2}} = 5 \times 3^{1/3} \]

### Step 5: Calculate the diagonal of the cube.
The diagonal \( d \) of a cube can be calculated using the formula:
\[ d = a \sqrt{3} \]
Substituting \( a \) from the previous step:
\[ d = (5 \times 3^{1/3}) \sqrt{3} = 5 \sqrt{3} \times 3^{1/3} = 5 \times 3^{1/2} = 5 \times \sqrt{3} \]

### Step 6: Calculate the numerical value of the diagonal.
Now we can calculate the numerical value of the diagonal:
\[ d = 5 \times \sqrt{3} \approx 5 \times 1.732 = 8.66 \, \text{cm} \]

### Final Answer:
Thus, the diagonal of the cube is approximately \( 8.66 \, \text{cm} \).

---