The total surface area of a cube is `"2904 cm"^(2)` . What is the volume of this cube ?

Options

A. `11748" cm"^(3)`
B. `10848" cm"^(3)`
C. `10748" cm"^(3)`
D. `10648" cm"^(3)`

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:
\[ \text{TSA} = 6A^2 \]
where \( A \) is the length of one side of the cube.

### Step 2: Set up the equation with the given total surface area
We know that the total surface area of the cube is \( 2904 \, \text{cm}^2 \). Therefore, we can set up the equation:
\[ 6A^2 = 2904 \]

### Step 3: Solve for \( A^2 \)
To find \( A^2 \), divide both sides of the equation by 6:
\[ A^2 = \frac{2904}{6} \]
Calculating the right side:
\[ A^2 = 484 \]

### Step 4: Find the value of \( A \)
Now, we need to find \( A \) by taking the square root of \( A^2 \):
\[ A = \sqrt{484} \]
Calculating the square root:
\[ A = 22 \, \text{cm} \]

### Step 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 \( A \):
\[ V = 22^3 \]
Calculating \( 22^3 \):
\[ V = 22 \times 22 \times 22 = 10648 \, \text{cm}^3 \]

### Final Answer
The volume of the cube is \( 10648 \, \text{cm}^3 \).

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