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(D) According to Torricelli's Law,the velocity of efflux is $v = \sqrt{2gh}$.The rate of change of volume of water in the vessel is given by $\frac{dV

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$2t_0$

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(D) According to Torricelli's Law,the velocity of efflux is $v = \sqrt{2gh}$.The rate of change of volume of water in the vessel is given by $\frac{dV}{dt} = -A_h v$,where $A_h$ is the area of the hole and $A$ is the cross-sectional area of the vessel.$A \frac{dh}{dt} = -A_h \sqrt{2gh}$Integrating this expression from height $H$ to $0$ in time $t_0$:$\int_{H}^{0} \frac{dh}{\sqrt{h}} = -\frac{A_h}{A} \sqrt{2g} \int_{0}^{t_0} dt$$2\sqrt{H} = \frac{A_h}{A} \sqrt{2g} \cdot t_0$This implies $t_0 \propto \sqrt{H}$.For a new height $H' = 4H$,the new time $t'$ is:$\frac{t'}{t_0} = \sqrt{\frac{H'}{H}} = \sqrt{\frac{4H}{H}} = 2$Therefore,$t' = 2t_0$.

$A$ cylindrical vessel filled with water up to the height $H$ becomes empty in time $t_0$ due to a small hole at the bottom of the vessel. If water is filled to a height $4H$,it will flow out in time:

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