A cylindrical vessel of cross-section \(A\) contains water to a height \(h\) . There is a hole in the bottom of radius \('a'\) . The time in which it will be emptied is

Initially at \(t = 0\,;\,h = h\) 

                \(t = t\,;h = 0\)

\(Then,\,A\left( { - \frac{{dh}}{{dt}}} \right) = \pi {a^2}.v\)

\(\left[ {Velocity\,of\,efflux\,of\,liquid\,v = \sqrt {2gh} } \right]\)

Integrating both sides

\(\int\limits_0^t {dt =  - \frac{A}{{\sqrt {2g} \pi {a^2}}}{{\int\limits_h^0 h }^{ - 1/2}}dh} \)

\(\left[ t \right]_0^t =  - \frac{A}{{\sqrt {2g} \pi {a^2}}} \cdot \left[ {\frac{{{h^{1/2}}}}{{1/2}}} \right]_h^0\)

\(t = \frac{{\sqrt 2 A}}{{\pi {a^2}}}\sqrt {\frac{h}{g}} \)