Find the volume of a cube whose total surface area is `384 cm^(2)`.

Answer

To find the volume of a cube whose total surface area is \(384 \, \text{cm}^2\), we can follow these steps:

### Step 1: Understand the formulas
The total surface area (TSA) of a cube is given by the formula:
\[ \text{TSA} = 6x^2 \]
where \(x\) is the length of one side of the cube.

### Step 2: Set up the equation
We know the total surface area is \(384 \, \text{cm}^2\). Therefore, we can set up the equation:
\[ 6x^2 = 384 \]

### Step 3: Solve for \(x^2\)
To find \(x^2\), we divide both sides of the equation by \(6\):
\[ x^2 = \frac{384}{6} \]
Calculating the right side:
\[ x^2 = 64 \]

### Step 4: Solve for \(x\)
Now, we take the square root of both sides to find \(x\):
\[ x = \sqrt{64} \]
Calculating the square root:
\[ x = 8 \, \text{cm} \]

### Step 5: Find the volume of the cube
The volume \(V\) of a cube is given by the formula:
\[ V = x^3 \]
Substituting the value of \(x\):
\[ V = 8^3 \]
Calculating \(8^3\):
\[ V = 512 \, \text{cm}^3 \]

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

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