How much water should be filled in a container of height `21 cm,` so that it appears half filled to the observer when viewed from the top of the container `(mu=4//3).`

Options

A. `8 cm`
B. `10.5 cm`
C. `12 cm`
D. `14 cm`

Explanation

To solve the problem of how much water should be filled in a container of height 21 cm so that it appears half-filled to an observer when viewed from the top, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Problem**:
We need to find the height of water (h) in a container of height 21 cm, such that when viewed from the top, it appears to be half-filled. The refractive index (μ) of water is given as \( \frac{4}{3} \).

2. **Define Apparent and Real Depth**:
- The real depth of the water is \( h \).
- The apparent depth, when viewed from the top, is \( 21 - h \) (since the total height of the container is 21 cm).

3. **Use the Formula for Refraction**:
The relationship between the real depth (h), apparent depth (21 - h), and the refractive index (μ) is given by the formula:
\[ \text{Apparent Depth} = \frac{\text{Real Depth}}{\mu} \]
Thus, we can write:
\[ 21 - h = \frac{h}{\mu} \]

4. **Substitute the Given Refractive Index**:
Substitute \( \mu = \frac{4}{3} \) into the equation:
\[ 21 - h = \frac{h}{\frac{4}{3}} \]
This can be rewritten as:
\[ 21 - h = \frac{3h}{4} \]

5. **Clear the Fraction**:
Multiply through by 4 to eliminate the fraction:
\[ 4(21 - h) = 3h \]
This simplifies to:
\[ 84 - 4h = 3h \]

6. **Combine Like Terms**:
Rearranging gives:
\[ 84 = 3h + 4h \]
Thus:
\[ 84 = 7h \]

7. **Solve for h**:
Divide both sides by 7:
\[ h = \frac{84}{7} = 12 \text{ cm} \]

### Conclusion:
The height of water that should be filled in the container is **12 cm**.

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