A small air bubble is trapped inside a transparent cube of size `12 cm`. When viewed from one of the vertical faces, the bubble apears to be at `5 cm`. From it. When viewed from opposite face, it appears at `3 cm` from it.

Options

A. The distance of the air bubble from the first face is 7.5 cm
B. The distance of the air bubble from the second face is 6 cm
C. Refractive index of the material of the prism is `(5)/(3)`
D. Refractive index of the material of the prism is 1.5

Answer

To solve the problem, we need to find the actual depth of the air bubble (denoted as \( T \)) and the refractive index of the material of the cube (denoted as \( \mu \)). We will use the concept of apparent depth in optics.

### Step-by-Step Solution:

1. **Understand the Geometry**:
- The cube has a side length of \( 12 \, \text{cm} \).
- When viewed from one vertical face, the bubble appears at \( 5 \, \text{cm} \) from that face.
- When viewed from the opposite vertical face, the bubble appears at \( 3 \, \text{cm} \) from that face.

2. **Set Up the Equations**:
- Let \( T \) be the actual depth of the bubble from the top face.
- The apparent depth when viewed from the first face is given by:
\[ \text{Apparent Depth} = \frac{T}{\mu} = 5 \quad \text{(Equation 1)} \]
- The apparent depth when viewed from the opposite face is given by:
\[ \text{Apparent Depth} = \frac{12 - T}{\mu} = 3 \quad \text{(Equation 2)} \]

3. **Express \( T \) in terms of \( \mu \)**:
- From Equation 1:
\[ T = 5\mu \quad \text{(Equation 3)} \]

4. **Substitute \( T \) into Equation 2**:
- Substitute Equation 3 into Equation 2:
\[ \frac{12 - 5\mu}{\mu} = 3 \]
- Multiply both sides by \( \mu \):
\[ 12 - 5\mu = 3\mu \]
- Rearranging gives:
\[ 12 = 8\mu \]
- Thus, we find:
\[ \mu = \frac{12}{8} = 1.5 \]

5. **Find the Actual Depth \( T \)**:
- Now substitute \( \mu = 1.5 \) back into Equation 3:
\[ T = 5 \times 1.5 = 7.5 \, \text{cm} \]

### Final Answers:
- The actual depth of the air bubble from the top face is \( 7.5 \, \text{cm} \).
- The refractive index of the material of the cube is \( 1.5 \).