1. Understand the situation

A vessel has total depth $d$ and is half filled with oil and half with water. So:

• thickness of oil layer $= \dfrac{d}{2}$
• thickness of water layer $= \dfrac{d}{2}$

We need the apparent depth when viewed from above.

2. Use apparent depth formula for each layer

For a layer of real thickness $t$ and refractive index $n$, when viewed from air, its apparent thickness is:

$t' = \frac{t}{n}$

Since the vessel contains two layers, the total apparent depth is the sum of the apparent thicknesses of both layers:

$d_{\text{app}} = \frac{d/2}{n_1} + \frac{d/2}{n_2}$

3. Simplify

$d_{\text{app}} = \frac{d}{2}\left(\frac{1}{n_1} + \frac{1}{n_2}\right)$

$d_{\text{app}} = \frac{d}{2}\left(\frac{n_1+n_2}{n_1 n_2}\right)$

$d_{\text{app}} = \frac{d(n_1+n_2)}{2n_1 n_2}$

4. Match with the options

This matches:

$\boxed{\frac{d(n_1+n_2)}{2n_1 n_2}}$

So the correct option is B.

5. Comparison with stored answer

Stored correct answer: B

Our derived answer: B

They agree.