A fish looking up through the water sees the outside world contained in circular horizon. If the R.I. of water `4/3` and tne fish is 12 cm below the surface, the radius of this circle in cm is-

Options

A. `36sqrt(7)`
B. `36/sqrt(7)`
C. `36sqrt(5)`
D. `4sqrt(5)`

Explanation

To solve the problem, we need to find the radius of the circular horizon that a fish sees when looking up through the water. We will use the formula for the radius of the circular region visible to the fish, which is derived from the refractive index of the medium.

### Step-by-Step Solution:

1. **Identify the Given Values**:
- Refractive Index (R.I.) of water, \( n = \frac{4}{3} \)
- Depth of the fish below the surface, \( h = 12 \, \text{cm} \)

2. **Use the Formula for the Radius**:
The formula for the radius \( R \) of the circular horizon is given by:
\[ R = \frac{h}{\sqrt{n^2 - 1}} \]

3. **Substitute the Values into the Formula**:
First, we need to calculate \( n^2 - 1 \):
\[ n^2 = \left(\frac{4}{3}\right)^2 = \frac{16}{9} \]
Therefore,
\[ n^2 - 1 = \frac{16}{9} - 1 = \frac{16}{9} - \frac{9}{9} = \frac{7}{9} \]

4. **Calculate the Square Root**:
Now, we take the square root of \( n^2 - 1 \):
\[ \sqrt{n^2 - 1} = \sqrt{\frac{7}{9}} = \frac{\sqrt{7}}{3} \]

5. **Substitute Back into the Radius Formula**:
Now we substitute back into the radius formula:
\[ R = \frac{12}{\sqrt{\frac{7}{9}}} = \frac{12}{\frac{\sqrt{7}}{3}} = 12 \times \frac{3}{\sqrt{7}} = \frac{36}{\sqrt{7}} \]

6. **Final Result**:
The radius of the circular horizon that the fish sees is:
\[ R = \frac{36}{\sqrt{7}} \, \text{cm} \]