A vessle of depth d is first filled to `( (2)/( 3))^(rd)` of its depth with a liquid of refractive index `( 3)/( 2)` and the rest of the vessel is filled with a liquid of refractive index `( 4)/( 3)`. What is the apparent depth of the inner surface of the bottom of the vessle.

Options

A. `( 5d)/( 16)`
B. `( 29d) /( 36)`
C. `(25 d)/( 36)`
D. `( 15 d )/( 16)`

Explanation

To find the apparent depth of the inner surface of the bottom of the vessel filled with two different liquids, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Depths and Refractive Indices**:
- The total depth of the vessel is \(d\).
- The first liquid (with refractive index \(\mu_1 = \frac{3}{2}\)) fills the vessel to a depth of \(\frac{2}{3}d\).
- The second liquid (with refractive index \(\mu_2 = \frac{4}{3}\)) fills the remaining depth of \(\frac{1}{3}d\).
...