A) \[\frac{2A}{\pi {{a}^{2}}}\sqrt{\frac{h}{g}}\]
B) \[\frac{\sqrt{2}A}{\pi {{a}^{2}}}\sqrt{\frac{h}{g}}\]
C) \[\frac{2\sqrt{2}A}{\pi {{a}^{2}}}\sqrt{\frac{h}{g}}\]
D) \[\frac{A}{\sqrt{2}\pi {{a}^{2}}}\sqrt{\frac{h}{g}}\]
Correct Answer: B
Solution :
Let the rate of falling water level be \[-\frac{dh}{dt}\] Initially at t = 0 ; h = h t = t ; h = 0 Then, \[A\left( -\frac{dh}{dt} \right)=\pi {{a}^{2}}.v\] \[dt=-\frac{A}{\pi {{a}^{2}}\sqrt{2gh}}dh\] [\[\because \]velocity of efflux of liquid \[v=\sqrt{2gh}\]] Integrating both sides \[\int\limits_{0}^{t}{dt}=-\frac{A}{\sqrt{2g}\pi {{a}^{2}}}\int\limits_{h}^{0}{{{h}^{-1/2}}}dh\] \[\left[ t \right]_{0}^{2}=-\frac{A}{\sqrt{2g}\pi {{a}^{2}}}.\left[ \frac{{{h}^{1/2}}}{1/2} \right]_{h}^{0}\] \[t=\frac{\sqrt{2}A}{\pi {{a}^{2}}}\sqrt{\frac{h}{g}}\]
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