A fish looking up through the water sees the outside world contained in a circular horizon. If the refractive index of water is `(4)/(3)` and the fish is 12 cm below the surface, the radius of this circle is cm is

Options

A. `(36)/(sqrt7)`
B. `36sqrt7`
C. `4sqrt5`
D. `36sqrt5`

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 concept of the critical angle and Snell's law to derive the radius.

### Step-by-Step Solution:

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

2. **Determine the Critical Angle:**
- According to Snell's law, when light travels from one medium to another, we have:
\[ n_1 \sin i = n_2 \sin r \]
- For the critical angle \( \theta_c \), the angle of refraction \( r \) is \( 90^\circ \). Thus, \( \sin 90^\circ = 1 \).
- Let \( n_1 = \mu_{water} = \frac{4}{3} \) and \( n_2 = \mu_{air} = 1 \).
- Applying Snell's law at the critical angle:
\[ \frac{4}{3} \sin \theta_c = 1 \cdot 1 \]
- Rearranging gives:
\[ \sin \theta_c = \frac{3}{4} \]

3. **Calculate \( \cos \theta_c \):**
- Using the identity \( \sin^2 \theta + \cos^2 \theta = 1 \):
\[ \cos^2 \theta_c = 1 - \sin^2 \theta_c = 1 - \left(\frac{3}{4}\right)^2 = 1 - \frac{9}{16} = \frac{7}{16} \]
- Therefore, \( \cos \theta_c = \sqrt{\frac{7}{16}} = \frac{\sqrt{7}}{4} \).

4. **Calculate \( \tan \theta_c \):**
- The tangent of the critical angle can be calculated as:
\[ \tan \theta_c = \frac{\sin \theta_c}{\cos \theta_c} = \frac{\frac{3}{4}}{\frac{\sqrt{7}}{4}} = \frac{3}{\sqrt{7}} \]

5. **Relate \( r \) and \( h \) using \( \tan \theta \):**
- In the right triangle formed by the fish, the radius \( r \) of the circular horizon and the depth \( h \):
\[ \tan \theta_c = \frac{r}{h} \]
- Substituting the values:
\[ \frac{3}{\sqrt{7}} = \frac{r}{12} \]
- Rearranging gives:
\[ r = 12 \cdot \frac{3}{\sqrt{7}} = \frac{36}{\sqrt{7}} \]

6. **Final Calculation:**
- The radius of the circular horizon is:
\[ r = \frac{36}{\sqrt{7}} \, \text{cm} \]

### Conclusion:
The radius of the circle that the fish sees is \( \frac{36}{\sqrt{7}} \) cm.