A transparent cube contains a small air bubble. Its apparent distance is 2 cm when seen through one face and 5 cm when seen through other face. If the refractive index of the material of the cube is 1.5, the real length of the edge of cube must be :

Question

Accepted Answer

To determine the real length of the edge of the cube, we need to consider the apparent distances of the air bubble when viewed from two different faces of the cube and the refractive index of the cube's material.

### Explanation:

1. **Understanding Apparent Distance:**
   - The apparent distance of an object in a medium is given by the formula:
     $$
     \text{Apparent Distance} = \frac{\text{Real Distance}}{\text{Refractive Index}}
     $$
   - Given that the refractive index ($ n $) of the cube is 1.5, we can use this formula to find the real distances from the two faces.

2. **Calculating Real Distances:**
   - For the first face, the apparent distance is 2 cm:
     $$
     2 = \frac{\text{Real Distance}_1}{1.5} \implies \text{Real Distance}_1 = 2 \times 1.5 = 3 \text{ cm}
     $$
   - For the second face, the apparent distance is 5 cm:
     $$
     5 = \frac{\text{Real Distance}_2}{1.5} \implies \text{Real Distance}_2 = 5 \times 1.5 = 7.5 \text{ cm}
     $$

3. **Determining the Edge Length of the Cube:**
   - The real distances from the two faces add up to the edge length of the cube:
     $$
     \text{Edge Length} = \text{Real Distance}_1 + \text{Real Distance}_2 = 3 \text{ cm} + 7.5 \text{ cm} = 10.5 \text{ cm}
     $$

### Justification for Option 4:

- **Option 4: $10.5 \text{ cm}$** is the correct answer because the calculated edge length of the cube, based on the given apparent distances and the refractive index, is 10.5 cm.

### Explanation of Incorrect Options:

- **Option 1: $\frac{14}{3} \text{ cm}$** is incorrect because this value does not match the calculated edge length of 10.5 cm.
- **Option 2: $7 \text{ cm}$** is incorrect because it is less than the sum of the real distances calculated from the apparent distances.
- **Option 3: $7.5 \text{ cm}$** is incorrect because it only accounts for one of the real distances, not the total edge length.

In conclusion, based on the calculations and the given refractive index, the real length of the edge of the cube is indeed 10.5 cm, making Option 4 the correct choice.