The surface area of a cube is `384 cm^(2)`. What is its volume?

Options

A. `1296 cm^(3)`
B. `648 cm^(3)`
C. `846 cm^(3)`
D. `512 cm^(3)`

Answer

To find the volume of a cube when the surface area is given, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the formula for the surface area of a cube**:
The surface area (SA) of a cube is given by the formula:
\[ SA = 6a^2 \]
where \( a \) is the length of one side of the cube.

2. **Set up the equation with the given surface area**:
We know the surface area is \( 384 \, cm^2 \). Therefore, we can set up the equation:
\[ 6a^2 = 384 \]

3. **Solve for \( a^2 \)**:
To isolate \( a^2 \), divide both sides of the equation by 6:
\[ a^2 = \frac{384}{6} \]
Simplifying this gives:
\[ a^2 = 64 \]

4. **Find the value of \( a \)**:
To find \( a \), take the square root of both sides:
\[ a = \sqrt{64} = 8 \, cm \]

5. **Calculate the volume of the cube**:
The volume (V) of a cube is given by the formula:
\[ V = a^3 \]
Substituting the value of \( a \):
\[ V = 8^3 \]
Calculating \( 8^3 \):
\[ V = 8 \times 8 \times 8 = 512 \, cm^3 \]

### Final Answer:
The volume of the cube is \( 512 \, cm^3 \).

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