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

Answer

To find the radius of the circular horizon that a fish sees when looking up through the water, we can follow these steps:

### Step 1: Understand the critical angle
The critical angle (\( \theta_c \)) is the angle of incidence above which total internal reflection occurs. For the fish to see the outside world, the angle of refraction must be 90 degrees.

### Step 2: Use Snell's Law
According to Snell's Law:
\[ \mu_1 \sin \theta_1 = \mu_2 \sin \theta_2 \]
Where:
- \( \mu_1 \) is the refractive index of water (\( \frac{4}{3} \))
- \( \mu_2 \) is the refractive index of air (approximately 1)
- \( \theta_1 \) is the angle of incidence (which is \( \theta_c \) in this case)
- \( \theta_2 \) is the angle of refraction (90 degrees)

Substituting the values:
\[ \frac{4}{3} \sin \theta_c = 1 \cdot \sin 90 \]
This simplifies to:
\[ \frac{4}{3} \sin \theta_c = 1 \]
Thus,
\[ \sin \theta_c = \frac{3}{4} \]

### Step 3: Find \( \tan \theta_c \)
Using the identity \( \sin^2 \theta + \cos^2 \theta = 1 \):
\[ \cos \theta_c = \sqrt{1 - \sin^2 \theta_c} = \sqrt{1 - \left(\frac{3}{4}\right)^2} = \sqrt{1 - \frac{9}{16}} = \sqrt{\frac{7}{16}} = \frac{\sqrt{7}}{4} \]
Now, we can find \( \tan \theta_c \):
\[ \tan \theta_c = \frac{\sin \theta_c}{\cos \theta_c} = \frac{\frac{3}{4}}{\frac{\sqrt{7}}{4}} = \frac{3}{\sqrt{7}} \]

### Step 4: Calculate the radius \( R \)
The radius of the circular horizon can be calculated using the relationship between the radius \( R \), the depth \( h \) of the fish, and \( \tan \theta_c \):
\[ R = h \tan \theta_c \]
Given that the fish is 12 cm below the surface (\( h = 12 \) cm):
\[ R = 12 \cdot \frac{3}{\sqrt{7}} = \frac{36}{\sqrt{7}} \text{ cm} \]

### Final Answer
Thus, the radius of the circular horizon that the fish sees is:
\[ R = \frac{36}{\sqrt{7}} \text{ cm} \]

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