A vessel of depth x is half filled with oil of refractive index `mu_(1)` and the other half is filled with water of refractive index `mu_(2)`. The apparent depth of the vessel when viewed above is

Options

A. `(x(mu_(1)+mu_(2)))/(2mu_(1)mu_(2))`
B. `(xmu_(1)mu_(2))/(2(mu_(1)+mu_(2)))`
C. `(xmu_(1)mu_(2))/((mu_(1)+mu_(2)))`
D. `(2x(mu_(1)+mu_(2)))/(mu_(1)mu_(2))`

Explanation

To find the apparent depth of the vessel when viewed from above, we need to consider the refraction of light as it passes through the different media (oil and water) with their respective refractive indices.

### Step-by-Step Solution:

1. **Identify the Depths and Refractive Indices:**
- The total depth of the vessel is \( X \).
- The depth of oil is \( \frac{X}{2} \) (since it is half-filled with oil).
- The depth of water is also \( \frac{X}{2} \).
- The refractive index of oil is \( \mu_1 \).
- The refractive index of water is \( \mu_2 \).

2. **Calculate the Apparent Depth of Oil:**
- The apparent depth \( d_1 \) of the oil layer can be calculated using the formula:
\[ d_1 = \frac{h_1}{\mu_1} \]
- Here, \( h_1 = \frac{X}{2} \) (depth of oil).
- Therefore,
\[ d_1 = \frac{\frac{X}{2}}{\mu_1} = \frac{X}{2\mu_1} \]

3. **Calculate the Apparent Depth of Water:**
- The apparent depth \( d_2 \) of the water layer can be calculated using the formula:
\[ d_2 = \frac{h_2}{\mu_2} \]
- Here, \( h_2 = \frac{X}{2} \) (depth of water).
- Therefore,
\[ d_2 = \frac{\frac{X}{2}}{\mu_2} = \frac{X}{2\mu_2} \]

4. **Combine the Apparent Depths:**
- The total apparent depth \( D \) of the vessel when viewed from above is the sum of the apparent depths of the oil and water:
\[ D = d_1 + d_2 \]
- Substituting the values we found:
\[ D = \frac{X}{2\mu_1} + \frac{X}{2\mu_2} \]

5. **Factor Out Common Terms:**
- We can factor out \( \frac{X}{2} \):
\[ D = \frac{X}{2} \left( \frac{1}{\mu_1} + \frac{1}{\mu_2} \right) \]

### Final Answer:
The apparent depth of the vessel when viewed from above is:
\[ D = \frac{X}{2} \left( \frac{1}{\mu_1} + \frac{1}{\mu_2} \right) \]