A pole of length 1.00 m stands half dipped in a swimming pool with water level 50.0 cm higher than the bed. The refractive index of water is 1.33 and sunlight is coming at an angle of 45° with the vertical. Find the length of the shadow of the pole on the bed.

Explanation

To solve the problem step by step, we will follow these steps:

### Step 1: Understand the Setup
We have a pole of length 1.00 m, which is half submerged in water. This means that 0.5 m of the pole is above the water and 0.5 m is below the water. The water level is 50 cm higher than the bed of the pool.

### Step 2: Identify the Angles
The sunlight is coming at an angle of 45° with the vertical. This means that the angle of incidence (I) with respect to the normal (vertical line) is 45°.

### Step 3: Apply Snell's Law
Using Snell's Law, we can find the angle of refraction (R) when light passes from air (n1 = 1) into water (n2 = 1.33). Snell's Law states:
\[ n_1 \sin I = n_2 \sin R \]
Substituting the values:
\[ 1 \cdot \sin(45°) = 1.33 \cdot \sin R \]
Since \(\sin(45°) = \frac{1}{\sqrt{2}}\):
\[ \frac{1}{\sqrt{2}} = 1.33 \cdot \sin R \]
Thus,
\[ \sin R = \frac{1}{\sqrt{2} \cdot 1.33} \]

### Step 4: Calculate \(\sin R\)
Calculating \(\sin R\):
\[ \sin R = \frac{1}{\sqrt{2} \cdot 1.33} \approx 0.53 \]

### Step 5: Find \(\cos R\)
Using the Pythagorean identity:
\[ \cos R = \sqrt{1 - \sin^2 R} \]
Calculating \(\cos R\):
\[ \cos R = \sqrt{1 - (0.53)^2} \approx 0.85 \]

### Step 6: Calculate \(\tan R\)
Now, we can find \(\tan R\):
\[ \tan R = \frac{\sin R}{\cos R} = \frac{0.53}{0.85} \approx 0.62 \]

### Step 7: Set Up the Triangle
In the triangle formed by the pole and its shadow on the bed, we can denote the length of the shadow on the bed as \(X\). The height of the pole above the water is 0.5 m. Thus, we can write:
\[ \tan R = \frac{X}{0.5} \]
Substituting the value of \(\tan R\):
\[ 0.62 = \frac{X}{0.5} \]

### Step 8: Solve for \(X\)
Now, solving for \(X\):
\[ X = 0.62 \cdot 0.5 = 0.31 \text{ m} \]

### Step 9: Calculate Total Length of Shadow
To find the total length of the shadow on the bed, we add the height of the pole above the water (0.5 m) to the length of the shadow \(X\):
\[ \text{Total Length of Shadow} = 0.5 + 0.31 = 0.81 \text{ m} \]

### Final Answer
The length of the shadow of the pole on the bed is **0.81 m** or **81.2 cm**.

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