As an air bubble rises from the bottom of a large water storage tank to free surface of water, the radius of the air bubble increases from 6 mm to 10 mm. The temperature of the water at the surface is `42^(@)C` and its bottom is `27^(@)C.` Find the depth of the water tank. (Take density of water `=1g cm^(-3), g=10 ms^(-2)`, 1 atmospheric pressure = 760 mm of Hg, density of mercury `=13.6 g cm^(-3)`

Explanation

To solve the problem, we need to apply the ideal gas law and the principles of hydrostatics. Here's a step-by-step solution:

### Step 1: Identify the known values
- Initial radius of the bubble, \( R_1 = 6 \, \text{mm} = 0.006 \, \text{m} \)
- Final radius of the bubble, \( R_2 = 10 \, \text{mm} = 0.01 \, \text{m} \)
- Temperature at the surface, \( T_2 = 42^\circ C = 315 \, \text{K} \)
- Temperature at the bottom, \( T_1 = 27^\circ C = 300 \, \text{K} \)
- Density of water, \( \rho = 1 \, \text{g/cm}^3 = 1000 \, \text{kg/m}^3 \)
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