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(A) Let $A$ be the base area of the cylinder and $A_0$ be the area of the orifice. The initial height of water above the orifice is $x$. The total hei

Vedclass · Accepted Answer

$80$

Vedclass · Answer

(A) Let $A$ be the base area of the cylinder and $A_0$ be the area of the orifice. The initial height of water above the orifice is $x$. The total height of water is $H = 1 \,m = 100 \,cm$.According to Torricelli's law, the velocity of efflux is $v = \sqrt{2gx}$.The rate of change of water level is given by $A \frac{dx}{dt} = -A_0 \sqrt{2gx}$.Separating variables and integrating from $x$ to $0$:$\int_{x}^{0} \frac{dx}{\sqrt{x}} = -\frac{A_0}{A} \sqrt{2g} \int_{0}^{t} dt$$2\sqrt{x} = \frac{A_0}{A} \sqrt{2g} \cdot t$Given $\frac{A}{A_0} = 100$, $t = 20 \,s$, and $g = 10 \,m/s^2$:$2\sqrt{x} = \frac{1}{100} \sqrt{2 	imes 10} 	imes 20$$2\sqrt{x} = \frac{1}{100} 	imes \sqrt{20} 	imes 20 = \frac{20}{100} 	imes 2\sqrt{5} = 0.4\sqrt{5}$$\sqrt{x} = 0.2\sqrt{5} \Rightarrow x = 0.04 	imes 5 = 0.2 \,m = 20 \,cm$.The height of the orifice from the base is $h = H - x = 100 \,cm - 20 \,cm = 80 \,cm$.

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