[Single-digit Integer] A swimming pool (Figure) is conical in shape of diameter 20 m. To measure the depth of the pool, a pipe line having cross sectional area π mm^2 has been introduced through the b

Question

[Single-digit Integer] A swimming pool (Figure) is conical in shape of diameter 20 m. To measure the depth of the pool, a pipe line having cross sectional area π mm^2 has been introduced through the base centre to the depth of the pool. Other end of the pipe line is connected with the gas cylinder. When gases are passed through the pipe line, it has been measured that the volume of gas with the change in bubble across the pipe line at the surface of the pool is 2π mm^3. Assume the temperature of gas is same as that of atmosphere and 1 atm pressure is equal to 10 m of water column. If the water holding capacity of the swimming pool is V m^3, then the value of 3V / (100 π) is

Prashant Kotian · Accepted Answer

Correct answer: 5. Using the pressure-volume relation for the bubble as it rises from depth h to the surface: P_bottom × V_bottom = P_surface × V_surface. P_surface = 10 m of water, V_surface = 2π mm^3. P_bottom = 10 + h. Using the pool's geometry, the depth is calculated to be h = 5 m. The volume of a cone is V = (1/3) π r^2 h. Since diameter is 20 m (r = 10 m): V = (1/3) π × 100 × 5 = 500π / 3 m^3. We need to find the value of 3V / (100 π) = 3 × (500π / 3) / (100π) = 5. Thus, the integer answer is 0005.

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