Find the volume of a cube whose total surface area is `486 cm^(2)`.

Answer

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

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

### Step 2: Set up the equation using the given total surface area.
Given that the total surface area is \(486 \, \text{cm}^2\), we can set up the equation:
\[ 6 \times \text{side}^2 = 486 \]

### Step 3: Solve for the side length.
To find the side length, we first divide both sides of the equation by 6:
\[ \text{side}^2 = \frac{486}{6} \]
Calculating the right side:
\[ \text{side}^2 = 81 \]

### Step 4: Take the square root to find the side length.
Now, we take the square root of both sides to find the side length:
\[ \text{side} = \sqrt{81} = 9 \, \text{cm} \]

### Step 5: Calculate the volume of the cube.
The volume (V) of a cube is given by the formula:
\[ V = \text{side}^3 \]
Substituting the value of the side:
\[ V = 9^3 = 729 \, \text{cm}^3 \]

### Final Answer:
The volume of the cube is \(729 \, \text{cm}^3\).

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