The longest rod that can be placed in cube shaped room is `15sqrt(3) cm`. What is the volume (in `cm^(3)`) of the cubical room ?

Options

A. 225
B. 3375
C. 1350
D. 625

Answer

To find the volume of the cubical room given that the longest rod that can be placed in it is \(15\sqrt{3}\) cm, we can follow these steps:

### Step 1: Understand the relationship between the diagonal and the side of the cube
The longest rod that can fit in a cube is the diagonal of the cube. The formula for the diagonal \(d\) of a cube with side length \(A\) is given by:
\[ d = A\sqrt{3} \]

### Step 2: Set up the equation
We know from the problem that the diagonal \(d\) is \(15\sqrt{3}\) cm. Therefore, we can set up the equation:
\[ A\sqrt{3} = 15\sqrt{3} \]

### Step 3: Solve for the side length \(A\)
To find \(A\), we can divide both sides of the equation by \(\sqrt{3}\):
\[ A = 15 \text{ cm} \]

### Step 4: Calculate the volume of the cube
The volume \(V\) of a cube is given by the formula:
\[ V = A^3 \]
Substituting the value of \(A\):
\[ V = 15^3 \]

### Step 5: Compute \(15^3\)
Calculating \(15^3\):
\[ 15^3 = 15 \times 15 \times 15 = 225 \times 15 = 3375 \text{ cm}^3 \]

### Final Answer
The volume of the cubical room is \(3375 \text{ cm}^3\).

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