A transparent cube of `15 cm` edge contains a small air bubble. Its apparent depth when viewed through one face is `6 cm` and when viewed through the opposite face is `4 cm`. Then the refractive index of the material of the cube is

Options

A. `2.0`
B. `2.5`
C. `1.6`
D. `1.5`

Explanation

To find the refractive index of the material of the cube, we can follow these steps:

### Step 1: Understand the Problem
We have a transparent cube with an edge length of 15 cm containing an air bubble. The apparent depths of the bubble when viewed through two opposite faces of the cube are given as 6 cm and 4 cm.

### Step 2: Define Variables
Let:
- \( x \) = actual depth of the bubble from the face where the apparent depth is 4 cm.
- Therefore, the actual depth from the opposite face (where the apparent depth is 6 cm) will be \( 15 - x \).

### Step 3: Use the Formula for Apparent Depth
The formula for apparent depth is given by:
\[ \text{Apparent Depth} = \frac{\text{Actual Depth}}{\mu} \]
where \( \mu \) is the refractive index of the material.

### Step 4: Set Up Equations
From the first face (apparent depth = 4 cm):
\[ 4 = \frac{x}{\mu} \quad \text{(1)} \]
From the opposite face (apparent depth = 6 cm):
\[ 6 = \frac{15 - x}{\mu} \quad \text{(2)} \]

### Step 5: Rearranging the Equations
From equation (1):
\[ \mu = \frac{x}{4} \]
From equation (2):
\[ \mu = \frac{15 - x}{6} \]

### Step 6: Set the Equations Equal to Each Other
Since both expressions equal \( \mu \):
\[ \frac{x}{4} = \frac{15 - x}{6} \]

### Step 7: Cross Multiply to Solve for \( x \)
Cross multiplying gives:
\[ 6x = 4(15 - x) \]
Expanding the right side:
\[ 6x = 60 - 4x \]

### Step 8: Combine Like Terms
Bringing all terms involving \( x \) to one side:
\[ 6x + 4x = 60 \]
\[ 10x = 60 \]

### Step 9: Solve for \( x \)
Dividing both sides by 10:
\[ x = 6 \text{ cm} \]

### Step 10: Substitute \( x \) Back to Find \( \mu \)
Now substituting \( x \) back into either equation for \( \mu \):
Using equation (1):
\[ \mu = \frac{6}{4} = 1.5 \]

### Final Answer
The refractive index of the material of the cube is \( \mu = 1.5 \).

---