An opaque cylindrical tank with an open top has a diameter of `3.00m` and is completely filled with water .When the setting sun reaches an angle of `37^(@)` above the horizon,sunlight ceases to illuminate any part of the bottom of the tank .How deep is the tank?

Options

A. 3 m
B. 4 m
C. 5 m
D. Not possible

Explanation

To solve the problem of finding the depth of the cylindrical tank, we will follow these steps:

### Step 1: Understand the Geometry of the Situation
We have a cylindrical tank with a diameter of 3.00 m. The sunlight is coming at an angle of 37° above the horizon. We need to determine the depth of the tank (h) at which sunlight no longer reaches the bottom.

### Step 2: Draw a Diagram
Draw a diagram of the cylindrical tank. Mark the diameter (3.00 m) and the angle of sunlight (37°). The sunlight will create a right triangle with the depth of the tank and the radius of the tank.

### Step 3: Calculate the Angle of Incidence
The angle of incidence (i) can be calculated as:
\[ i = 90° - 37° = 53° \]

### Step 4: Apply Snell's Law
Using Snell's Law, we have:
\[ \frac{\sin(i)}{\sin(r)} = n \]
where \( n \) is the refractive index of water (approximately 1.33).

Substituting the known values:
\[ \sin(53°) = 0.79 \]
Thus, we can write:
\[ \frac{0.79}{\sin(r)} = 1.33 \]

### Step 5: Solve for the Angle of Refraction
Rearranging gives:
\[ \sin(r) = \frac{0.79}{1.33} \]
Calculating this gives:
\[ \sin(r) \approx 0.60 \]

Now, find the angle of refraction (r):
\[ r = \sin^{-1}(0.60) \approx 36.87° \]

### Step 6: Set Up the Right Triangle
In the right triangle formed by the radius (1.5 m, which is half of the diameter), the depth (h), and the angle of refraction (r):
\[ \tan(r) = \frac{\text{opposite}}{\text{adjacent}} = \frac{3 \text{ m}}{h} \]

### Step 7: Calculate the Depth of the Tank
From the tangent function:
\[ \tan(36.87°) = \frac{3}{h} \]
Calculating \( \tan(36.87°) \) gives approximately 0.75. Therefore:
\[ 0.75 = \frac{3}{h} \]

Rearranging gives:
\[ h = \frac{3}{0.75} = 4 \text{ m} \]

### Conclusion
The depth of the tank is 4 meters.

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