A large open tank has two holes in the walls. One is a square hole of side $L$ at a depth $y$ from the top and other is a circular hole of radius $R$ at a depth $4 y$ from the top. When the tank is completely filled with water the quantities of water flowing out per second from both the holes are the same. then, $R$ is equal to:

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$2 π L$

$\frac{L}{\sqrt{2 π}}$

$L$

$\frac{L}{2 π}$

answer is B.

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Detailed Solution

Velocity of efflux when the hole is at depth $h$ , $v = \sqrt{2 g h}$

Rate of flow of water from square hole,

$Q_{1} = a_{1} v_{1} = L^{2} \sqrt{2 g y}$

Rate of flow of water from circular hole,

$Q_{2} = a_{2} v_{2} = π R^{2} \sqrt{2 g (4 y)}$

According to problem,

$L^{2} \sqrt{2 g y} = π R^{2} \sqrt{2 g (4 y)}$

or $R = \frac{L}{\sqrt{2 π}}$

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