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(B) The velocity of efflux from a hole at depth $h$ is given by Torricelli's law: $v = \sqrt{2gh}$.The volume of water flowing out per second (flow ra

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$r\sqrt{2\pi}$

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(B) The velocity of efflux from a hole at depth $h$ is given by Torricelli's law: $v = \sqrt{2gh}$.The volume of water flowing out per second (flow rate) is $Q = A \cdot v$,where $A$ is the area of the hole.For hole $P$ (square of side $a$ at depth $2h$): $A_P = a^2$ and $v_P = \sqrt{2g(2h)} = \sqrt{4gh} = 2\sqrt{gh}$.Thus,flow rate $Q_P = a^2 \cdot 2\sqrt{gh}$.For hole $Q$ (circle of radius $r$ at depth $8h$): $A_Q = \pi r^2$ and $v_Q = \sqrt{2g(8h)} = \sqrt{16gh} = 4\sqrt{gh}$.Thus,flow rate $Q_Q = \pi r^2 \cdot 4\sqrt{gh}$.Given $Q_P = Q_Q$,we have: $a^2 \cdot 2\sqrt{gh} = \pi r^2 \cdot 4\sqrt{gh}$.$a^2 \cdot 2 = 4\pi r^2 \implies a^2 = 2\pi r^2$.Taking the square root of both sides: $a = r\sqrt{2\pi}$.

$A$ vessel completely filled with water has two holes $P$ and $Q$ at depths $2h$ and $8h$ from the top,respectively. Hole $P$ is a square of side $a$ and hole $Q$ is a circle of radius $r$. If the volume of water flowing out per second from both holes is the same,then the side $a$ of hole $P$ is:

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