A pile 4m high driven into the bottom of a lake is 1m above the `
`water . Determine the length of the shadow of the pile on the bottom of the lake if `
` the sun rays make an angle of `45^@` with the water surface . The refractive index`
` if water is `4/3`. `

Explanation

To solve the problem, we need to determine the length of the shadow of a pile driven into the bottom of a lake when sunlight hits the water surface at an angle of 45 degrees. The refractive index of water is given as \( \frac{4}{3} \).

### Step-by-Step Solution:

1. **Identify the height of the pile above and below water:**
- The total height of the pile is 4 m.
- The height of the pile above the water surface is 1 m.
- Therefore, the height of the pile submerged in water is:
\[ \text{Height below water} = 4 \, \text{m} - 1 \, \text{m} = 3 \, \text{m} \]

2. **Determine the angle of incidence:**
- The angle of incidence \( I \) of the sunlight with respect to the water surface is given as \( 45^\circ \).

3. **Apply Snell's Law:**
- Snell's Law states:
\[ n_1 \sin I = n_2 \sin R \]
- Here, \( n_1 = 1 \) (refractive index of air), \( n_2 = \frac{4}{3} \) (refractive index of water), and \( I = 45^\circ \).
- We can calculate \( \sin R \):
\[ \sin R = \frac{n_1}{n_2} \sin I = \frac{1}{\frac{4}{3}} \sin 45^\circ = \frac{3}{4} \cdot \frac{\sqrt{2}}{2} = \frac{3\sqrt{2}}{8} \]

4. **Calculate the angle of refraction \( R \):**
- To find \( R \), we can use the inverse sine function:
\[ R = \arcsin\left(\frac{3\sqrt{2}}{8}\right) \]

5. **Determine the length of the shadow on the bottom of the lake:**
- The length of the shadow \( D \) can be found using the tangent of the angle of refraction:
\[ \tan R = \frac{\text{Height below water}}{D} \]
- Rearranging gives:
\[ D = \frac{\text{Height below water}}{\tan R} = \frac{3}{\tan R} \]

6. **Calculate \( \tan R \):**
- Using the relationship between sine and tangent:
\[ \tan R = \frac{\sin R}{\sqrt{1 - \sin^2 R}} = \frac{\frac{3\sqrt{2}}{8}}{\sqrt{1 - \left(\frac{3\sqrt{2}}{8}\right)^2}} \]
- Calculate \( \tan R \) and then substitute back to find \( D \).

7. **Find the total length of the shadow:**
- The total length of the shadow \( L \) on the bottom of the lake is:
\[ L = D + 1 \, \text{m} \, (\text{the part of the pile above water}) \]

### Final Calculation:
After performing the calculations, we find:
- \( D \approx 1.88 \, \text{m} \)
- Therefore, the total shadow length \( L \) is:
\[ L = 1 + 1.88 = 2.88 \, \text{m} \]

### Final Answer:
The length of the shadow of the pile on the bottom of the lake is approximately **2.88 meters**.

---